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a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.
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%I #41 Oct 17 2024 21:31:02

%S 1,1,5,73,2069,95401,6487445,610093513,75796724309,12020754177001,

%T 2369364111428885,568128719132038153,162835627057766030549,

%U 54975855375379966645801,21593185551426744571090325,9762238510837560633366673993,5033241437347149354018370856789

%N a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

%C Number of digraphs with loops, with labeled vertices and labeled arcs, with n arcs and with no vertex of indegree 0 or outdegree 0, cf. A121936, A122418, A122399. - _Vladeta Jovovic_, Sep 06 2006

%C Chromatic invariant of the complete bipartite graph K_{n+1,n+1}. - _Eric W. Weisstein_, Jul 11 2011

%C Generally, for p >= 1, Sum_{k=0..n} (k!*StirlingS2(n,k))^p is asymptotic to n^(p*n+1/2) * sqrt(Pi/(2*p*(1-log(2))^(p-1))) / (exp(p*n) * log(2)^(p*n+1)). - _Vaclav Kotesovec_, May 10 2014

%H Alois P. Heinz, <a href="/A048144/b048144.txt">Table of n, a(n) for n = 0..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticInvariant.html">Chromatic Invariant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%F E.g.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(exp(j*x)-1)^n. a(n) = Sum_{k=0..n} Stirling2(n,k)*k!*A104602(k). - _Vladeta Jovovic_, Mar 25 2006

%F a(n) ~ sqrt(Pi/(1-log(2))) * n^(2*n+1/2) / (2*exp(2*n) * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, May 09 2014

%F E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n * exp(-n*x). - _Paul D. Hanna_, Mar 26 2018

%F E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n * exp(-n*(n+1)*x). - _Paul D. Hanna_, Mar 26 2018

%F a(n) = A272644(2n,n). - _Alois P. Heinz_, Oct 17 2024

%t Table[Sum[(k!)^2*StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, May 07 2014 *)

%o (PARI) a(n) = sum(k=0, n, k!^2*stirling(n, k, 2)^2); \\ _Michel Marcus_, Mar 07 2020

%Y Cf. A000670, A242280, A212084.

%Y Cf. A120732, A104602, A272644.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_