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A122852
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Row sums of number triangle A122851.
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9
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1, 1, 2, 3, 6, 11, 24, 51, 122, 291, 756, 1979, 5526, 15627, 46496, 140451, 442194, 1414931, 4687212, 15785451, 54764846, 193129659, 698978136, 2570480147, 9672977706, 36967490691, 144232455524, 571177352091, 2304843053382, 9434493132011, 39289892366736
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.
G.f.: 1/(1-x-x^2/(1-x^2/(1-x-2x^2/(1-2x^2/(1-x-3x^2/(1-3x^2/(1-x-4x^2/(1-4x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)
D-finite with recurrence -2*a(n) + 3*a(n-1) + (n-1)*a(n-2) + (-n+1)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012. Proof in [Han 2019]
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 + 1/8) * n^((n+1)/2) / 2^(n/2+1) * (1 + 37/(48*sqrt(2*n))). - Vaclav Kotesovec, Feb 08 2014
a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - Seiichi Manyama, Nov 19 2022
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MATHEMATICA
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Table[Sum[Binomial[n-k, k]*k!, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 08 2014 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(k, n-k)*(n-k)!); \\ Michel Marcus, Sep 02 2020
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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