A002288 -id:A002288

A002611

Glaisher's function V(n).
(Formerly M3235 N1305)

+10
2

0, 1, 4, -4, -32, -16, 56, 80, 192, 98, -740, -704, 96, -224, 2440, 3520, -2624, -351, -780, -10632, 2688, 2960, -9496, 18176, 14208, -3934, 12552, -9856, -24608, -9760, -2720, -25344, -35520, 31106, 34160, 62844, 84576, 3120, -21880, -82272, 27520, -96768, -237316, 130240, -92832, 37984, 305296, -183296, 37632, 208803

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

REFERENCES

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 320). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..50.

Index entries for sequences mentioned by Glaisher

FORMULA

a(n) = Sum_{k=1..floor(n/2)} A004018(n - 2*k) * A002288(k). - Sean A. Irvine, Mar 04 2019

CROSSREFS

Cf. A002288, A004018.

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited and signs added by N. J. A. Sloane, Nov 26 2018

More terms from Sean A. Irvine, Mar 04 2019

STATUS

approved

A002614

Glaisher's function theta(n) (18 squares version).
(Formerly M4459 N1890)

+10
2

0, -7, 128, -975, 4608, -16340, 48384, -124303, 281600, -583746, 1146240, -2125108, 3691008, -6151880, 10055424, -15914895, 24136704, -35748899, 52583040, -75877938, 105994240, -145580124, 200279808, -272040500, 359036928, -468767690, 615599360

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

REFERENCES

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 349). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..500

Index entries for sequences mentioned by Glaisher

FORMULA

a(n) = (A321546(n) - A002288(n)) / 17. - Sean A. Irvine, Mar 04 2019

CROSSREFS

Cf. A002288, A321546.

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited and signs added by N. J. A. Sloane, Nov 26 2018

More terms from Sean A. Irvine, Mar 04 2019

STATUS

approved

A034433

Expansion of q^(-3) * (eta(q) * eta(q^8))^8 in powers of q.

+10
2

1, -8, 20, 0, -70, 64, 56, 0, -133, -96, 148, 0, 670, -512, -968, 0, 1077, 1680, -2064, 0, -2098, 768, 4400, 0, -1766, -8128, 7044, 0, 744, 4096, -4760, 0, -9780, 16344, -6652, 0, 7894, -13440, -10320, 0, 41923, -8736, -16780, 0, -5892, -6144, 14560, 0, -27886, -11056, 55940

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

Euler transform of period 8 sequence [ -8, -8, -8, -8, -8, -8, -8, -16, ...]. - Michael Somos, Nov 11 2007

a(4*n+3) = 0.

EXAMPLE

q^3 - 8*q^4 + 20*q^5 - 70*q^7 + 64*q^8 + 56*q^9 - 133*q^11 - 96*q^12 + ...

MATHEMATICA

QP = QPochhammer; s = (QP[q]*QP[q^8])^8 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^8 + A) )^8, n))} /* Michael Somos, Nov 11 2007 */

CROSSREFS

-8 * A002288(n) = a(4*n-3).

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved

A216711

Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.

+10
2

1, 8, 12, -64, -210, 96, 1016, 512, -2043, -1680, 1092, -768, 1382, 8128, -2520, -4096, 14706, -16344, -39940, 13440, 12192, 8736, 68712, 6144, -34025, 11056, -50760, -65024, -102570, -20160, 227552, 32768, 13104, 117648, -213360, 130752, 160526, -319520

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.

a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).

a(n) = -(-1)^n * A002288(n). Convolution square of A134461.

EXAMPLE

G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};

(Magma) A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */

CROSSREFS

Cf. A002288, A134461.

KEYWORD

sign,mult,look,changed

AUTHOR

Michael Somos, Apr 10 2013

STATUS

approved

A002272

Theta series of 32-dimensional Quebbemann lattice Q_32.

+10
1

1, 0, 0, 261120, 18947520, 535818240, 8320327680, 83347937280, 622558664640, 3614759362560, 17694184734720, 73337844372480, 272615629589760, 898646461378560, 2752654757806080, 7687895624386560

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..15.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

G. Nebe and N. J. A. Sloane, Home page for this lattice.

H.-G. Quebbemann, Lattices with theta-functions for G(sqrt(2)) and linear codes, J. Algebra, 105 (1987), 443-450.

H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190-202.

FORMULA

G.f.: b(x)^8 - 192*b(x)^4*d(x) + 576*d(x)^2 where b(x) is the g.f. of A004011 and d(x) is the g.f. of A002288. - Sean A. Irvine, Jul 26 2020

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A002610

Glaisher's function H'(4n+1) (18 squares version).
(Formerly M4051 N1681)

+10
1

0, 1, -6, -3, 82, -84, -444, 769, 1110, -2643, -860, 2901, -1176, 6277, 1170, -21315, -2308, 14244, 29442, 15540, -58194, -13338, -31886, 4080, 176682, -70715, -51240, 81489, -135728, 13137, -205350, 58826, 355974, 16380, 530932, -457944, -938748, 140329, 99462, 317157

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

REFERENCES

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 312). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..39.

Sean A. Irvine, Java program (github)

Index entries for sequences mentioned by Glaisher

FORMULA

See eqn. top of page 312 in Glaisher, where Theta(n) is A002288(n). - Sean A. Irvine, Mar 03 2019

CROSSREFS

Cf. A002288.

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited and signs added by N. J. A. Sloane, Nov 26 2018

More terms from Sean A. Irvine, Mar 03 2019

STATUS

approved

A002615

Glaisher's function T_1(n).
(Formerly M5076 N2196)

+10
1

19, -145, 100, 2191, -8598, 14516, -29080, 114575, -320417, 615666, -1125492, 2139700, -3664750, 5997448, -10103304, 15992719, -23857290, 36059435, -53341900, 75622578, -105762592, 145414140, -198974280, 271923764, -359683403, 468557626

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

Glaisher table on p. 349 apparently has typos, a(5) = -8592, a(10) = 615566. - Sean A. Irvine, Mar 04 2019

REFERENCES

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 349). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..500

Index entries for sequences mentioned by Glaisher

FORMULA

a(n) = 19 * A002288(n) - A002614(n). - Sean A. Irvine, Mar 04 2019

CROSSREFS

Cf. A002288, A002614.

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited and signs added by N. J. A. Sloane, Nov 26 2018

a(5) and a(10) corrected and more terms from Sean A. Irvine, Mar 04 2019

STATUS

approved

A173763

Expansion of (eta(q^2)^7 / eta(q^4)^2)^4 + 16 * q * (eta(q)^2 * eta(q^2) * eta(q^4)^2)^4 in powers of q.

+10
1

1, 16, -156, 256, 870, -2496, -952, 4096, 4653, 13920, -56148, -39936, 178094, -15232, -135720, 65536, -247662, 74448, 315380, 222720, 148512, -898368, 204504, -638976, -1196225, 2849504, 2344680, -243712, -3840450, -2171520, -1309408, 1048576, 8759088, -3962592, -828240, 1191168, 4307078

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q * (psi(q)^3 * phi(-q)^2)^4 * ((phi(q) / psi(q))^4 + 16 * q * (psi(q) / phi(q))^4) in powers of q where phi(), psi() are Ramanujan theta functions.

a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = a(p) * a(p^(e-1)) - p^9 * a(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 32 (t / i)^10 f(t) where q = exp(2 Pi i t).

EXAMPLE

G.f. = q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 - 2496*q^6 - 952*q^7 + 4096*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^7 / QPochhammer[ q^4]^2)^4 + 16 q^2 (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^4, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / eta(x^4 + A)^2)^4 + 16 * x * (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^2)^4, n))};

(PARI) q='q+O('q^99); Vec((eta(q^2)^7/eta(q^4)^2)^4+16*q*(eta(q)^2*eta(q^2)*eta(q^4)^2)^4) \\ Altug Alkan, Apr 18 2018

(Sage) CuspForms( Gamma1(2), 10, prec=50).0; # Michael Somos, May 28 2013

(Magma) Basis( CuspForms( Gamma1(2), 10), 50) [1]; /* Michael Somos, May 27 2014 */

CROSSREFS

Cf. A002288.

KEYWORD

sign,mult,changed

AUTHOR

Michael Somos, Feb 23 2010

EXTENSIONS

Name corrected by Altug Alkan, Apr 18 2018

STATUS

approved

A319456

a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

+10
1

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..29.

FORMULA

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.

a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).

MATHEMATICA

Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]

Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]

Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]