A395780 - OEIS

%I #13 May 06 2026 08:48:19

%S 1,16,180,1744,15554,131616,1074072,8536240,66489687,509765952,

%T 3858791664,28905439248,214635119470,1581949708352,11585423108592,

%U 84377281155168,611551322732687,4413481365001680,31730521183547172,227349945847349952,1623986957273609304

%N Expansion of (Sum_{k>=0} binomial(3*k+1,k) * x^k)^4.

%F a(n) = Sum_{i,j,k,l >= 0 and i+j+k+l=n} binomial(3*i+1,i) * binomial(3*j+1,j) * binomial(3*k+1,k) * binomial(3*l+1,l).

%F G.f.: B(x)^4 where B(x) is the g.f. of A045721.

%F a(n) = Sum_{k=0..n} 2^k * binomial(k+2,2) * binomial(3*n+7,n-k).

%F a(n) = Sum_{k=0..n} 3^k * binomial(k+2,2) * binomial(3*n+4-k,n-k).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+16) * 2^k * binomial(k+3,3) * binomial(3*n+7,n-1-k).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (2*k+16) * 3^k * binomial(k+3,3) * binomial(3*n+3-k,n-1-k).

%o (PARI) a(n) = sum(k=0, n, 2^k*binomial(k+2, 2)*binomial(3*n+7, n-k));

%Y Cf. A045721, A062236, A395779.

%Y Cf. A045894.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, May 05 2026