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A336502 Partial sums of A057003.

%I A336502 #39 Nov 20 2021 08:14:36
%S A336502 1,7,127,5167,365527,39435607,6006997207,1226103906007,
%T A336502 322796982334807,106460296033918807,42980408446129381207,
%U A336502 20846482682939051365207,11959807608801430284133207,8010447502346968140207973207,6193994326661240674349352805207,5476021766725276671842502543205207
%N A336502 Partial sums of A057003.
%C A336502 Inspired by doubly triangular numbers (A002817).
%H A336502 Seiichi Manyama, <a href="/A336502/b336502.txt">Table of n, a(n) for n = 1..226</a>
%F A336502 a(n) = Sum_{i=1..n} Product_{j=T(i-1)+1..T(i)} j where T(n) is n-th triangular number.
%F A336502 a(n) = A227364(T(n)) where T(n) is n-th triangular number.
%F A336502 a(n) ~ n^(2*n) / 2^n. - _Vaclav Kotesovec_, Nov 20 2021
%e A336502 a(2) = 1 + 2*3 = 7.
%e A336502 a(3) = 1 + 2*3 + 4*5*6 = 127.
%e A336502 a(4) = 1 + 2*3 + 4*5*6 + 7*8*9*10 = 5167.
%t A336502 Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* _Amiram Eldar_, Jul 23 2020 *)
%o A336502 (PARI) {a(n) = sum(i=1, n, prod(j=(i-1)*i/2+1, i*(i+1)/2, j))}
%Y A336502 Cf. A000217, A002817, A007489, A057003, A227364, A336513.
%K A336502 nonn,easy
%O A336502 1,2
%A A336502 _Seiichi Manyama_, Jul 23 2020