A395783 - OEIS

%I #15 May 06 2026 08:48:06

%S 1,15,159,1460,12405,100449,787390,6030240,45386460,337036980,

%T 2476230876,18036064140,130431953357,937603882245,6705639192060,

%U 47748355535600,338708562711465,2394692570004255,16881072760978035,118691594469460080,832588258306289580,5828191882336453020

%N Expansion of (Sum_{k>=0} binomial(3*k+2,k) * x^k)^3.

%F a(n) = Sum_{i,j,k >= 0 and i+j+k=n} binomial(3*i+2,i) * binomial(3*j+2,j) * binomial(3*k+2,k).

%F G.f.: (B(x)/x)^3 where B(x) is the g.f. of A025174.

%F a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(3*n+8,n-k).

%F a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(3*n+6-k,n-k).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+15) * 2^k * binomial(k+2,2) * binomial(3*n+8,n-1-k).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (2*k+15) * 3^k * binomial(k+2,2) * binomial(3*n+5-k,n-1-k).

%o (PARI) a(n) = sum(k=0, n, (k+1)*2^k*binomial(3*n+8, n-k));

%Y Cf. A025174, A386617, A395785.

%Y Cf. A378483, A395779.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, May 05 2026