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Partial sums of A057003.
2

%I #39 Nov 20 2021 08:14:36

%S 1,7,127,5167,365527,39435607,6006997207,1226103906007,

%T 322796982334807,106460296033918807,42980408446129381207,

%U 20846482682939051365207,11959807608801430284133207,8010447502346968140207973207,6193994326661240674349352805207,5476021766725276671842502543205207

%N Partial sums of A057003.

%C Inspired by doubly triangular numbers (A002817).

%H Seiichi Manyama, <a href="/A336502/b336502.txt">Table of n, a(n) for n = 1..226</a>

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

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

%F a(n) ~ n^(2*n) / 2^n. - _Vaclav Kotesovec_, Nov 20 2021

%e a(2) = 1 + 2*3 = 7.

%e a(3) = 1 + 2*3 + 4*5*6 = 127.

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

%t Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* _Amiram Eldar_, Jul 23 2020 *)

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

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

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jul 23 2020