A187154 -id:A187154

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A115671

Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

+10
6

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369

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

OFFSET

0,4

COMMENTS

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

Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Reinhard Zumkeller)

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).

G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.

Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.

Expansion of f(q^6, q^10) / f(-q, -q^3) = (f(q^22, q^26) - q^2 * f(q^10, q^38)) / f(-q, -q^2) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 32 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].

Given g.f. A(x), then B(x) = (2*A(x) - 1)^2 = g.f. A007096 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 1 + u^2 - 2 * u * v^2.

G.f. (1 + sqrt( theta_3(x) / theta_4(x))) / 2 = (Sum_{k} x^(8*k^2 - 2*k)) / (Sum_{k} (-x)^(2*k^2 - k)) = (Sum_{k} x^(24*n^2 + 2*n) - x^(24*n^2 + 14*n + 2)) / (Product_{k>0} 1 - x^k).

2 * a(n) = A080054(n) unless n = 0. a(2*n + 2) = A208851(n). a(2*n + 1) = A187154(n). a(n + 1) = A208856(n).

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...

a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.

a(6) = 6 since 6 = 5 + 1 = 4 + 1 + 1 = 3 + 3 = 3 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

(Haskell)

a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0, 2, 12, 14, 16, 18, 20, 30]]

where p _ 0 = 1

p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Mar 03 2012

CROSSREFS

Cf. A080054, A187154, A208851, A208856, A218171.

KEYWORD

nonn,changed

AUTHOR

Michael Somos, Jan 29 2006

STATUS

approved

A210063

Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

+10
6

1, -2, 4, -8, 15, -26, 44, -72, 114, -178, 272, -408, 605, -884, 1276, -1824, 2580, -3616, 5028, -6936, 9498, -12922, 17468, -23472, 31369, -41700, 55156, -72616, 95172, -124202, 161436, -209016, 269616, -346562, 443952, -566856, 721530, -915642, 1158608

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

OFFSET

0,2

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5 in powers of q.

Euler transform of period 8 sequence [ -2, 3, -2, 2, -2, 3, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A210030.

a(n) = (-1)^n * A187154(n). Convolution inverse of A208589.

a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (16*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

EXAMPLE

1 - 2*x + 4*x^2 - 8*x^3 + 15*x^4 - 26*x^5 + 44*x^6 - 72*x^7 + 114*x^8 + ...

q - 2*q^3 + 4*q^5 - 8*q^7 + 15*q^9 - 26*q^11 + 44*q^13 - 72*q^15 + 114*q^17 + ...

MATHEMATICA

CoefficientList[Series[x^(-1/2) * EllipticTheta[2, 0, x^2] / (2*EllipticTheta[3, 0, x]), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 17 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5, n))}

CROSSREFS

Cf. A187154, A208589, A210030.

KEYWORD

sign,changed

AUTHOR

Michael Somos, Mar 16 2012

STATUS

approved

A208856

Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

+10
3

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369, 36189

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

OFFSET

0,3

COMMENTS

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

Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.

Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].

a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).

EXAMPLE

1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...

a(5) = 6 since 5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.

a(6) = 8 since 5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.

MATHEMATICA

A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n, 0, 50}] (* G. C. Greubel, Jun 19 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

CROSSREFS

Cf. A080054, A115671, A187154, A208851.

KEYWORD

nonn,changed

AUTHOR

Michael Somos, Mar 02 2012

STATUS

approved

A093085

Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

+10
2

1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968

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

OFFSET

0,2

COMMENTS

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

eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].

Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.

G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.

EXAMPLE

G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...

G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...

MATHEMATICA

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

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}

(PARI) {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}

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

CROSSREFS

Cf. A029838, A029839, A083365, A131124, A187154.

KEYWORD

sign,changed

AUTHOR

Michael Somos, Mar 20 2004, Oct 22 2007

STATUS

approved

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