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

%I A266083 #11 Feb 16 2025 08:33:28
%S A266083 1,1,3,12,54,280,1695,11886,95116,856080,8560845,94169350,1130032266,
%T A266083 14690419536,205665873595,3084988104030,49359809664600,
%U A266083 839116764298336,15104101757370201,286977933390033990,5739558667800679990,120530732023814280000,2651676104523914160231
%N A266083 a(n) = Sum_{k = 0..n - 1} (a(n - 1) + k) for n>0, a(0) = 1.
%H A266083 G. C. Greubel, <a href="/A266083/b266083.txt">Table of n, a(n) for n = 0..320</a>
%H A266083 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>
%F A266083 a(n) = (2*n! + exp(1)*n*(n - 1)*Gamma(n - 1, 1))/2, where Gamma(a, x) is the incomplete gamma function.
%F A266083 a(n + 1) - a(n)*(n + 1) = A000217(n).
%F A266083 a(n) = n*a(n-1) + binomial(n,2). - _G. C. Greubel_, Dec 22 2015
%e A266083 a(0) = 1;
%e A266083 a(1) = 1 + 0 = 1;
%e A266083 a(2) = 1 + 0 + 1 + 1 = 3;
%e A266083 a(3) = 3 + 0 + 3 + 1 + 3 + 2 = 12;
%e A266083 a(4) = 12 + 0 + 12 + 1 + 12 + 2 + 12 + 3 = 54;
%e A266083 a(5) = 54 + 0 + 54 + 1 + 54 + 2 + 54 + 3 + 54 + 4 = 280, etc.
%t A266083 Table[(2 n! + Exp[1] n (n - 1) Gamma[n - 1, 1])/2, {n, 0, 22}]
%t A266083 RecurrenceTable[{a[n] == n*a[n - 1] + Binomial[n, 2], a[0] == 1}, a, {n, 0, 20}] (* _G. C. Greubel_, Dec 22 2015 *)
%o A266083 (PARI) a(n) = (2*n! + exp(1)*n*(n-1)*incgam(n-1, 1))\/2
%Y A266083 Cf. A000217, A038155 (for a(0) = 0).
%K A266083 nonn
%O A266083 0,3
%A A266083 _Ilya Gutkovskiy_, Dec 21 2015