A052702 -id:A052702

Search: a052702 -id:a052702

Displaying 1-4 of 4 results found.

page 1

A160705

Hankel transform of A052702.

+20
2

0, 0, 0, 0, 1, 1, -1, -4, -4, 5, 9, 9, -14, -16, -16, 30, 25, 25, -55, -36, -36, 91, 49, 49, -140, -64, -64, 204, 81, 81, -285, -100, -100, 385, 121, 121, -506, -144, -144, 650, 169, 169, -819, -196, -196, 1015, 225, 225, -1240, -256, -256

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

OFFSET

0,8

COMMENTS

a(n+5) is the Hankel transform of A052702(n+4).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (0,0,-4,0,0,-6,0,0,-4,0,0,-1).

FORMULA

G.f.: x^4*(1-x)*(1+x+x^2)*(x^4+x^3-x^2+x+1)/( (1+x)^4*(x^2-x+1)^4 ).

a(n) = -4*a(n-3) -6*a(n-6) -4*a(n-9) -a(n-12).

MATHEMATICA

LinearRecurrence[{0, 0, -4, 0, 0, -6, 0, 0, -4, 0, 0, -1}, {0, 0, 0, 0, 1, 1, -1, -4, -4, 5, 9, 9}, 50] (* G. C. Greubel, May 02 2018 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(x^4*(1-x)*(1+x+x^2)*(x^4+x^3-x^2+x+1)/( (1+x)^4*(x^2-x+1)^4 ))) \\ G. C. Greubel, May 02 2018

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0] cat Coefficients(R!(x^4*(1-x)*(1+x+x^2)*(x^4+x^3-x^2+x+1)/( (1+x)^4*(x^2-x+1)^4 ))); // G. C. Greubel, May 02 2018

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 24 2009

STATUS

approved

A160706

Hankel transform of A052702(n+1).

+20
2

0, 0, 0, 1, 0, -1, -2, 0, 2, 3, 0, -3, -4, 0, 4, 5, 0, -5, -6, 0, 6, 7, 0, -7, -8, 0, 8, 9, 0, -9, -10, 0, 10, 11, 0, -11, -12, 0, 12, 13, 0, -13, -14, 0, 14, 15, 0, -15, -16, 0, 16, 17, 0, -17, -18, 0, 18, 19, 0, -19, -20

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

OFFSET

0,7

COMMENTS

a(n+5) is the Hankel transform of A052702(n+3).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (1,-1,-1,1,-1).

FORMULA

G.f.: x^3*(1-x^2)/(1 + x^3)^2.

MATHEMATICA

LinearRecurrence[{1, -1, -1, 1, -1}, {0, 0, 0, 1, 0}, 100] (* G. C. Greubel, May 02 2018 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0, 0], Vec(x^3*(1-x^2)/(1 + x^3)^2)) \\ G. C. Greubel, May 02 2018

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0] cat Coefficients(R!(x^3*(1-x^2)/(1 + x^3)^2)); // G. C. Greubel, May 02 2018

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 24 2009

STATUS

approved

A361229

G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

+10
1

1, 0, 0, 0, 1, 2, 3, 4, 7, 14, 27, 48, 84, 152, 284, 532, 987, 1826, 3401, 6384, 12024, 22656, 42728, 80780, 153151, 290970, 553601, 1054688, 2012373, 3845646, 7359345, 14100692, 27048061, 51941850, 99855389, 192163904, 370159216, 713672568, 1377168108, 2659729380

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

OFFSET

0,6

LINKS

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

FORMULA

G.f.: A(x) = 2*(1-x) / (1-x+sqrt((1-x)^2-4*x^4)).

a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1).

D-finite with recurrence (n+4)*a(n) +(-3*n-7)*a(n-1) +(3*n+2)*a(n-2) +(-n+1)*a(n-3) +4*(-n+2)*a(n-4) +4*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023

MAPLE

A361229 := proc(n)

add(binomial(n-2*k-1, n-4*k) * binomial(2*k, k) / (k+1), k=0..floor(n/4)) ;

end proc:

seq(A361229(n), n=0..70) ; # R. J. Mathar, Dec 04 2023

PROG

(PARI) a(n) = sum(k=0, n\4, binomial(n-2*k-1, n-4*k)*binomial(2*k, k)/(k+1));

CROSSREFS

Partial sums give A023426.

Cf. A002026, A006319, A052702.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 15 2023

STATUS

approved

A052724

A simple context-free grammar in a labeled universe: a(n) = A052743(n)-A052723(n), n>1.

+10
0

0, 0, 0, 0, 24, 240, 2160, 30240, 524160, 9434880, 188697600, 4311014400, 108254361600, 2939153817600, 86568043161600, 2753962219008000, 93838712647680000, 3409619685728256000, 131735241369059328000

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

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..18.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 680

FORMULA

E.g.f.: (1/2)/x^2*(1-x-(1-2*x+x^2-4*x^3)^(1/2))-(1/2)/x*(1-x-(1-2*x+x^2-4*x^3)^(1/2))-x

Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, a(6)=2160, a(7)=30240, (38*n^4+120*n^3-48-4*n+130*n^2+4*n^5)*a(n) +(-193*n^2-52*n^3-302*n-5*n^4-168)*a(n+1) +(96+29*n^2+92*n+3*n^3)*a(n+2) +(-52-3*n^2-25*n)*a(n+3) +(n+6)*a(n+4)=0, a(5)=240}

a(n) = n!*A052702(n). - R. J. Mathar, Oct 18 2013

MAPLE

spec := [S, {B=Prod(Z, C), C=Union(B, S, Z), S=Prod(B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

seq(n!*add(binomial(n-k-2, 2*k-1)*binomial(2*k, k)/(k+1), k=0..n-2), n=0..20); # Mark van Hoeij, May 12 2013

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

page 1

Search completed in 0.005 seconds