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A266083
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a(n) = Sum_{k = 0..n - 1} (a(n - 1) + k) for n>0, a(0) = 1.
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2
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1, 1, 3, 12, 54, 280, 1695, 11886, 95116, 856080, 8560845, 94169350, 1130032266, 14690419536, 205665873595, 3084988104030, 49359809664600, 839116764298336, 15104101757370201, 286977933390033990, 5739558667800679990, 120530732023814280000, 2651676104523914160231
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (2*n! + exp(1)*n*(n - 1)*Gamma(n - 1, 1))/2, where Gamma(a, x) is the incomplete gamma function.
a(n + 1) - a(n)*(n + 1) = A000217(n).
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EXAMPLE
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a(0) = 1;
a(1) = 1 + 0 = 1;
a(2) = 1 + 0 + 1 + 1 = 3;
a(3) = 3 + 0 + 3 + 1 + 3 + 2 = 12;
a(4) = 12 + 0 + 12 + 1 + 12 + 2 + 12 + 3 = 54;
a(5) = 54 + 0 + 54 + 1 + 54 + 2 + 54 + 3 + 54 + 4 = 280, etc.
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MATHEMATICA
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Table[(2 n! + Exp[1] n (n - 1) Gamma[n - 1, 1])/2, {n, 0, 22}]
RecurrenceTable[{a[n] == n*a[n - 1] + Binomial[n, 2], a[0] == 1}, a, {n, 0, 20}] (* G. C. Greubel, Dec 22 2015 *)
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PROG
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(PARI) a(n) = (2*n! + exp(1)*n*(n-1)*incgam(n-1, 1))\/2
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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