A266083 a(n) = Sum_{k = 0..n - 1} (a(n - 1) + k) for n>0, a(0) = 1.

1, 1, 3, 12, 54, 280, 1695, 11886, 95116, 856080, 8560845, 94169350, 1130032266, 14690419536, 205665873595, 3084988104030, 49359809664600, 839116764298336, 15104101757370201, 286977933390033990, 5739558667800679990, 120530732023814280000, 2651676104523914160231

Offset: 0

Author

Ilya Gutkovskiy, Dec 21 2015

Keywords

• nonn

Examples

			a(0) = 1;
a(1) = 1 + 0 = 1;
a(2) = 1 + 0 + 1 + 1 = 3;
a(3) = 3 + 0 + 3 + 1 + 3 + 2 = 12;
a(4) = 12 + 0 + 12 + 1 + 12 + 2 + 12 + 3 = 54;
a(5) = 54 + 0 + 54 + 1 + 54 + 2 + 54 + 3 + 54 + 4 = 280, etc.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..320
• Eric Weisstein's World of Mathematics, Incomplete Gamma Function

Crossrefs

Cf. A000217, A038155 (for a(0) = 0).

Programs

Mathematica

Table[(2 n! + Exp[1] n (n - 1) Gamma[n - 1, 1])/2, {n, 0, 22}]
RecurrenceTable[{a[n] == n*a[n - 1] + Binomial[n, 2], a[0] == 1}, a, {n, 0, 20}] (* G. C. Greubel, Dec 22 2015 *)

PARI

a(n) = (2*n! + exp(1)*n*(n-1)*incgam(n-1, 1))\/2

Formula

a(n) = (2*n! + exp(1)*n*(n - 1)*Gamma(n - 1, 1))/2, where Gamma(a, x) is the incomplete gamma function.

a(n + 1) - a(n)*(n + 1) = A000217(n).

a(n) = n*a(n-1) + binomial(n,2). - G. C. Greubel, Dec 22 2015