A395782 - OEIS

%I #12 May 06 2026 08:47:58

%S 1,20,294,3804,45881,529308,5921820,64781900,696650580,7391238736,

%T 77568523822,806770798500,8328009400398,85417717917288,

%U 871280511426936,8844707030389164,89408619343690220,900444396245910768,9038422593653829688,90455699063988431856,902846885411835421209

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

%F a(n) = Sum_{i,j,k,l >= 0 and i+j+k+l=n} binomial(4*i+1,i) * binomial(4*j+1,j) * binomial(4*k+1,k) * binomial(4*l+1,l).

%F G.f.: B(x)^4 where B(x) is the g.f. of A052203.

%F a(n) = Sum_{k=0..n} 3^k * binomial(k+2,2) * binomial(4*n+7,n-k).

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

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (4*k+20) * 3^k * binomial(k+3,3) * binomial(4*n+7,n-1-k).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+20) * 4^k * binomial(k+3,3) * binomial(4*n+3-k,n-1-k).

%o (PARI) a(n) = sum(k=0, n, 3^k*binomial(k+2, 2)*binomial(4*n+7, n-k));

%Y Cf. A052203, A308523, A395781.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, May 05 2026