A052702 -id:A052702 - cp's OEIS frontend

A160705 Hankel transform of A052702.

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

Offset: 0

Paul Barry, May 24 2009

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

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).

m:=50; R:=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

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 *)

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

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).

A160706 Hankel transform of A052702(n+1).

Original entry on oeis.org

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

Offset: 0

Paul Barry, May 24 2009

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

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).

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

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

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

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

A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A023426.

Cf. A002026, A006319, A052702.

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

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

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

A052724 A simple context-free grammar in a labeled universe: a(n) = A052743(n)-A052723(n), n>1.

Original entry on oeis.org

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

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 680

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

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

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A160705 Hankel transform of A052702.

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A160706 Hankel transform of A052702(n+1).

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A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

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A052724 A simple context-free grammar in a labeled universe: a(n) = A052743(n)-A052723(n), n>1.

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