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A122420
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Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.
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4
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1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = (1/n!)*Sum_{k=0..n} |Stirling1(n,k)|*A122418(k). G.f.: A(x/(1-x)) where A(x) is g.f. for A122419.
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014
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MAPLE
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A122418 := proc(n) option remember ; add( combinat[stirling2](n, k)*(k-1)^n*k!, k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n, k))*A122418(k), k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ", A122420(n)) ; od ; # R. J. Mathar, May 18 2007
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MATHEMATICA
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Table[1/n!*Sum[Abs[StirlingS1[n, k]]*Sum[(m-1)^k*m!*StirlingS2[k, m], {m, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2014 *)
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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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