A337001 - OEIS

%I #17 Jan 13 2024 06:46:30

%S 0,1,10,57,292,1585,9726,68425,547912,4931937,49320370,542525401,

%T 6510306540,84633987217,1184875823782,17773137360105,284370197765776,

%U 4834293362023105,87017280516421722,1653328329812019577,33066566596240399540,694397898521048399601

%N a(n) = n! * Sum_{k=0..n} k^3 / k!.

%C Exponential convolution of cubes (A000578) and factorial numbers (A000142).

%H Seiichi Manyama, <a href="/A337001/b337001.txt">Table of n, a(n) for n = 0..449</a>

%F E.g.f.: x * (1 + 3*x + x^2) * exp(x) / (1 - x).

%F a(0) = 0; a(n) = n * (n^2 + a(n-1)).

%F a(n) ~ 5*exp(1)*n!. - _Vaclav Kotesovec_, Jan 13 2024

%t Table[n! Sum[k^3/k!, {k, 0, n}], {n, 0, 21}]

%t nmax = 21; CoefficientList[Series[x (1 + 3 x + x^2) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 0; a[n_] := a[n] = n (n^2 + a[n - 1]); Table[a[n], {n, 0, 21}]

%o (PARI) a(n) = n! * sum(k=0, n, k^3/k!); \\ _Michel Marcus_, Aug 12 2020

%Y Cf. A000142, A000522, A000578, A007526, A030297, A256016, A337002.

%Y Row sums of A121682.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 10 2020