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A264889 Partial sums of hyperfactorials (A002109).

%I A264889 #25 Feb 16 2025 08:33:27
%S A264889 1,2,6,114,27762,86427762,4031164827762,3319770429936027762,
%T A264889 55696441261496986915227762,21577941278638297470665013744027762,
%U A264889 215779412250996503370318565758665013744027762,61564384586850833363801728392684283449726665013744027762
%N A264889 Partial sums of hyperfactorials (A002109).
%H A264889 G. C. Greubel, <a href="/A264889/b264889.txt">Table of n, a(n) for n = 0..37</a>
%H A264889 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%H A264889 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%F A264889 a(n) = Sum_{k = 0..n} A002109(k).
%F A264889 a(n) = Sum_{k = 0..n} (k!)^k/Barnes G-Function(k + 1).
%e A264889 a(0) = 1;
%e A264889 a(1) = 1 + 1^1 = 2;
%e A264889 a(2) = 1 + 1^1 + 1^1*2^2 = 6;
%e A264889 a(3) = 1 + 1^1 + 1^1*2^2 + 1^1*2^2*3^3 = 114;
%e A264889 a(4) = 1 + 1^1 + 1^1*2^2 + 1^1*2^2*3^3 + 1^1*2^2*3^3*4^4 = 27762, etc.
%t A264889 Table[Sum[Hyperfactorial[k], {k, 0, n}], {n, 0, 11}]
%t A264889 Accumulate[Hyperfactorial[Range[0,15]]] (* _Harvey P. Dale_, Sep 22 2021 *)
%o A264889 (PARI) a(n) = sum(k=0, n, prod(j=2, k, j^j)); \\ _Altug Alkan_, Nov 27 2015
%Y A264889 Cf. A002109, A007489, A152690.
%K A264889 nonn
%O A264889 0,2
%A A264889 _Ilya Gutkovskiy_, Nov 27 2015