A118968 a(4n+k) = (k+1)*binomial(5n+k,n)/(4n+k+1), k=0..3.

1, 1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 80, 136, 204, 285, 665, 1155, 1771, 2530, 5980, 10530, 16380, 23751, 56637, 100688, 158224, 231880, 556512, 996336, 1577532, 2330445, 5620485, 10116873, 16112057, 23950355, 57985070, 104819165

Offset

0,6

Comments

Row sums of Riordan array (1,x(1-x^4))^(-1).

Links

Table of n, a(n) for n=0..38.

F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Formula

a(4n)=A002294(n), a(4n+1)=A118969(n), a(4n+2)=A118970(n), a(4n+3)=A118971(n).

G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x)*A(I*x)*A(-I*x). - Paul D. Hanna, Jun 04 2012

G.f. satisfies: A(x) = 1 + x*A(x)*G(x^4) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294. - Paul D. Hanna, Jun 04 2012

From Robert A. Russell, Mar 14 2024: (Start)

G.f.: G(z^4) + z*G(z^4)^2 + z^2*G(z^4)^3 + z^3*G(z^4)^4, where G(z) = 1 + z*G(z)^5 is the g.f. for A002294.

G.f.: E(1)(t*E(5)(t^4)) (fifth entry in Table 3), where E(d)(t) is defined in formula 3 of Hering link. (End)

Mathematica

Table[k=Mod[n, 4]; (k+1)Binomial[(5n-k)/4, (n-k)/4]/(n+1), {n, 0, 40}] (* Robert A. Russell, Mar 14 2024 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*A^2*subst(A, x, -x)*subst(A, x, I*x)*subst(A, x, -I*x)); polcoeff(A, n)} \\ Paul D. Hanna, Jun 04 2012
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*exp(sum(m=1, n\4, 4*polcoeff(log(A+x*O(x^n)), 4*m)*x^(4*m))+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Jun 04 2012

Crossrefs

Cf. A124753, A002294, A118969, A118970, A118971.

Sequence in context: A052418 A051800 A121432 * A073528 A116067 A063685

Adjacent sequences: A118965 A118966 A118967 * A118969 A118970 A118971

Keyword

easy,nonn

Author

Paul Barry, May 07 2006

Status

approved