A391649 - OEIS

%I #14 Dec 20 2025 02:55:32

%S 1,1,3,14,75,438,2704,17356,114661,774514,5324812,37137379,262112002,

%T 1868610856,13436074970,97328988039,709609323325,5203171782012,

%U 38345109874752,283864224950635,2109947626538162,15740670860816827,117819995601231131,884569446129847355

%N G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)^3)^2).

%H Vincenzo Librandi, <a href="/A391649/b391649.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1 - x*B(x)^2), where B(x) is the g.f. of A367239.

%F a(n) = Sum_{k=0..n} binomial(3*n-2*k+1,k) * binomial(n+k-1,n-k)/(3*n-2*k+1).

%t Table[Sum[Binomial[3*n-2*k+1,k]*Binomial[n+k-1,n-k]/(3*n-2*k+1),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 20 2025 *)

%o (PARI) a(n, s=2, t=1, u=3) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));

%o (Magma) [&+[Binomial(3*n-2*k+1,k)*Binomial(n+k-1, n-k)/(3*n-2*k+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 20 2025

%Y Cf. A367239.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 15 2025