A336502 Partial sums of A057003.

1, 7, 127, 5167, 365527, 39435607, 6006997207, 1226103906007, 322796982334807, 106460296033918807, 42980408446129381207, 20846482682939051365207, 11959807608801430284133207, 8010447502346968140207973207, 6193994326661240674349352805207, 5476021766725276671842502543205207

Offset: 1

Author

Seiichi Manyama, Jul 23 2020

Keywords

• nonn
• easy

Comments

Inspired by doubly triangular numbers (A002817).

Examples

			a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3 + 4*5*6 = 127.
a(4) = 1 + 2*3 + 4*5*6 + 7*8*9*10 = 5167.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..226

Crossrefs

Cf. A000217, A002817, A007489, A057003, A227364, A336513.

Programs

Mathematica

Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* Amiram Eldar, Jul 23 2020 *)

PARI

{a(n) = sum(i=1, n, prod(j=(i-1)*i/2+1, i*(i+1)/2, j))}

Formula

a(n) = Sum_{i=1..n} Product_{j=T(i-1)+1..T(i)} j where T(n) is n-th triangular number.

a(n) = A227364(T(n)) where T(n) is n-th triangular number.

a(n) ~ n^(2*n) / 2^n. - Vaclav Kotesovec, Nov 20 2021