cp's OEIS Frontend

A072678 Generalized Bell numbers B_{4,2}.

%I A072678 #21 Sep 05 2022 21:47:31
%S A072678 1,21,1045,93289,12975561,2581284541,693347907421,241253367679185,
%T A072678 105394372192969489,56410454014314490981,36271084122927079387941,
%U A072678 27567930377271475039277881,24435533594428382909107147225
%N A072678 Generalized Bell numbers B_{4,2}.
%H A072678 Robert Israel, <a href="/A072678/b072678.txt">Table of n, a(n) for n = 1..220</a>
%H A072678 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>, arXiv:quant-phys/0402027, 2004.
%H A072678 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.
%H A072678 M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.
%F A072678 a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
%F A072678 a(n) = sum(A090438(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-2), 2), j=1..n), k=1..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
%F A072678 8*n*(2*n-1)*(2*n+1)*(n+1)^2*(n+3)*(n+2)*a(n)+(2*(n+1))*(8*n^3+32*n^2+42*n+13)*a(n+1)*(n+3)*(n+2)-(8*n^2+38*n+51)*(n+3)*(n+2)*a(n+2)+(n+3)*(n+2)*a(n+3) = 0. - _Robert Israel_, May 23 2016
%F A072678 a(n) = A052852(2*n-1). - _Mark van Hoeij_, Sep 05 2022
%p A072678 f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
%p A072678 map(f, [$1..30]); # _Robert Israel_, May 23 2016
%t A072678 a[n_] := n*(2n-1)!*Hypergeometric1F1[2-2n, 3, -1]; Array[a, 30] (* _Jean-François Alcover_, Sep 01 2016 *)
%Y A072678 Cf. A090439 (alternating row sums of A090438).
%K A072678 nonn
%O A072678 1,2
%A A072678 _Karol A. Penson_, Jul 01 2002
%E A072678 Edited by _Wolfdieter Lang_, Dec 23 2003