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A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

%I A048144 #51 Apr 24 2025 06:35:11
%S A048144 1,1,5,73,2069,95401,6487445,610093513,75796724309,12020754177001,
%T A048144 2369364111428885,568128719132038153,162835627057766030549,
%U A048144 54975855375379966645801,21593185551426744571090325,9762238510837560633366673993,5033241437347149354018370856789
%N A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.
%C A048144 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 A048144 Chromatic invariant of the complete bipartite graph K_{n+1,n+1}. - _Eric W. Weisstein_, Jul 11 2011
%C A048144 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 A048144 Alois P. Heinz, <a href="/A048144/b048144.txt">Table of n, a(n) for n = 0..100</a>
%H A048144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticInvariant.html">Chromatic Invariant</a>
%H A048144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%F A048144 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 A048144 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 A048144 E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n * exp(-n*x). - _Paul D. Hanna_, Mar 26 2018
%F A048144 E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n * exp(-n*(n+1)*x). - _Paul D. Hanna_, Mar 26 2018
%F A048144 a(n) = A272644(2n,n). - _Alois P. Heinz_, Oct 17 2024
%F A048144 a(n) = A371761(n, n). - _Peter Luschny_, Nov 20 2024
%F A048144 a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y)). - _Ilya Gutkovskiy_, Apr 24 2025
%p A048144 a := proc(n) local A, j; A := proc(n, k) option remember; if n = 0 then n^k else add(binomial(k + `if`(j>0, 1, 0), j+1) * A(n-1, k-j), j = 0..k) fi end: A(n,n) end:
%p A048144 seq(a(n), n = 0..16);  # _Peter Luschny_, Nov 20 2024
%t A048144 Table[Sum[(k!)^2*StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, May 07 2014 *)
%o A048144 (PARI) a(n) = sum(k=0, n, k!^2*stirling(n, k, 2)^2); \\ _Michel Marcus_, Mar 07 2020
%o A048144 (Python)
%o A048144 from functools import cache
%o A048144 from math import comb as binomial
%o A048144 @cache
%o A048144 def A(n, k): return int(k == 0) if n == 0 else sum(binomial(k + int(j > 0), j + 1) * A(n - 1, k - j) for j in range(k + 1))
%o A048144 a = lambda n: A(n, n)
%o A048144 print([a(n) for n in range(17)])  # _Peter Luschny_, Nov 20 2024
%Y A048144 Cf. A000670, A242280, A212084, A120732, A104602, A272644, A371761.
%K A048144 nonn,easy
%O A048144 0,3
%A A048144 _N. J. A. Sloane_