A122418 -id:A122418 - cp's OEIS frontend

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

1, 1, 5, 73, 2069, 95401, 6487445, 610093513, 75796724309, 12020754177001, 2369364111428885, 568128719132038153, 162835627057766030549, 54975855375379966645801, 21593185551426744571090325, 9762238510837560633366673993, 5033241437347149354018370856789

Offset: 0

N. J. A. Sloane

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
Chromatic invariant of the complete bipartite graph K_{n+1,n+1}. - Eric W. Weisstein, Jul 11 2011
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

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Cf. A000670, A242280, A212084, A120732, A104602, A272644, A371761.

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:
seq(a(n), n = 0..16);  # Peter Luschny, Nov 20 2024

Table[Sum[(k!)^2*StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

a(n) = sum(k=0, n, k!^2*stirling(n, k, 2)^2); \\ Michel Marcus, Mar 07 2020

from functools import cache
from math import comb as binomial
@cache
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))
a = lambda n: A(n, n)
print([a(n) for n in range(17)])  # Peter Luschny, Nov 20 2024

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
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
E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n * exp(-n*x). - Paul D. Hanna, Mar 26 2018
E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n * exp(-n*(n+1)*x). - Paul D. Hanna, Mar 26 2018
a(n) = A272644(2n,n). - Alois P. Heinz, Oct 17 2024
a(n) = A371761(n, n). - Peter Luschny, Nov 20 2024
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y)). - Ilya Gutkovskiy, Apr 24 2025

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

Original entry on oeis.org

3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7

Offset: 1

Vaclav Kotesovec, Aug 09 2018

			3.161088653865428813830172202588132491726382774188556341627278...

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.

Cf. A301584, A301585, A301586, A303056, A303057, A303654.

r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]

r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 8, 93, 1354, 23900, 496244, 11855700, 320428318, 9667220397, 322072882348, 11744421711587, 465270864839688, 19899234175413257, 913836170567749048, 44849438199960187278, 2342666125012348876152

Offset: 0

Vladeta Jovovic, Sep 03 2006

Cf. A122420, A122400.

A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122419 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122419(n)) ; od ; # R. J. Mathar, May 18 2007

nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A122418(k).
G.f.: Sum_{n>=0} ((1+x)^(n-1) - 1)^n.
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 = 0.08904589343883135100956914504938... . - Vaclav Kotesovec, May 07 2014

More terms from R. J. Mathar, May 18 2007

A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068

Offset: 0

Vladeta Jovovic, Sep 03 2006

Cf. A104209.

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

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 *)

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

More terms from R. J. Mathar, May 18 2007

A121936 Number of digraphs with labeled vertices and labeled arcs, with n arcs and with no vertex of indegree 0 or outdegree 0.

Original entry on oeis.org

1, 0, 2, 18, 518, 23610, 1600982, 150451098, 18694217558, 2966151496410, 584994048653462, 140357794553191578, 40253455215544778198, 13598018000464234802010, 5343837921922909297592342

Offset: 0

Vladeta Jovovic, Sep 03 2006

Cf. A122399, A122418.

E.g.f.: Sum(Sum((-1)^(n-k)*binomial(n,k)*(exp((k-1)*x)-1)^k*(exp(k*x)-1)^(n-k),k=0..n),n=0..infinity).

More terms from Max Alekseyev, Jul 29 2009

cp's OEIS Frontend

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

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A317855 Decimal expansion of a constant related to the asymptotics of A122400.

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A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.

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A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

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A121936 Number of digraphs with labeled vertices and labeled arcs, with n arcs and with no vertex of indegree 0 or outdegree 0.

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