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

%I A118968 #33 Jul 20 2025 15:02:22
%S A118968 1,1,1,1,1,2,3,4,5,11,18,26,35,80,136,204,285,665,1155,1771,2530,5980,
%T A118968 10530,16380,23751,56637,100688,158224,231880,556512,996336,1577532,
%U A118968 2330445,5620485,10116873,16112057,23950355,57985070,104819165,167710664,250543370,608462470
%N A118968 a(4n+k) = (k+1)*binomial(5n+k,n)/(4n+k+1), k=0..3.
%C A118968 Row sums of Riordan array (1,x(1-x^4))^(-1).
%H A118968 F. Hering et al., <a href="http://dx.doi.org/10.1016/0012-365X(82)90121-2">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203-217.
%F A118968 a(4n) = A002294(n), a(4n+1) = A118969(n), a(4n+2) = A118970(n), a(4n+3) = A118971(n).
%F A118968 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x)*A(I*x)*A(-I*x). - _Paul D. Hanna_, Jun 04 2012
%F A118968 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
%F A118968 From _Robert A. Russell_, Mar 14 2024: (Start)
%F A118968 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.
%F A118968 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)
%F A118968 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} a(4*k) * a(n-1-4*k). - _Seiichi Manyama_, Jul 07 2025
%t A118968 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 *)
%o A118968 (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
%o A118968 (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
%o A118968 (PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
%o A118968 a(n) = apr(n\4, 5, n%4+1); \\ _Seiichi Manyama_, Jul 20 2025
%Y A118968 Cf. A124753, A002294, A118969, A118970, A118971.
%K A118968 easy,nonn
%O A118968 0,6
%A A118968 _Paul Barry_, May 07 2006