cp's OEIS Frontend

A078738 Generalized Bell numbers B_{3,2}(n).

%I A078738 #32 Apr 21 2025 08:37:24
%S A078738 1,13,355,16333,1121881,106708921,13354028563,2118817455385,
%T A078738 414426460442833,97746679844312581,27311169061720393411,
%U A078738 8908525371578726747173,3350963996380181114090665,1438463413778071631322236593,698374517715612292764726380851
%N A078738 Generalized Bell numbers B_{3,2}(n).
%H A078738 Vaclav Kotesovec, <a href="/A078738/b078738.txt">Table of n, a(n) for n = 1..240</a>
%H A078738 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%H A078738 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-ph/0402027, 2004.
%H A078738 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 A078738 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 A078738 a(n) = Sum_{k=2..2*n} A078740(n, k) = Sum_{k=1..infinity} (1/k!)*Product_{j=1..n}(fallfac(k+(j-1)*(3-2), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
%F A078738 a(n) = Sum_{k>=0} ((n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))/exp(1), n>=1. From eq.(40) of the Blasiak et al. reference. [corrected by _Vaclav Kotesovec_, Jul 27 2018]
%F A078738 E.g.f. for a(n)/n! with a(0)=(exp(1)-1)/exp(1) added: Sum_{k>=0} (hypergeom([k+2, k+1], [1], z)/(k+2)!)/exp(1). From eq. (41) of the Blasiak et al. reference.
%t A078738 a[n_] := (n+1)*n!^2*Sum[(-1)^k*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!), {k, 2, 2n}]; Array[a, 13] (* _Jean-François Alcover_, Sep 01 2015 *)
%t A078738 Table[Sum[(n + k)!*(n + k + 1)!/(k!*(k + 1)!*(k + 2)!), {k, 0, Infinity}]/E, {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 27 2018 *)
%o A078738 (PARI) nmax = 20; p = floor(3*nmax*log(nmax)); default(realprecision, p);
%o A078738 for(n=1, nmax, print1(round(exp(-1)*suminf(k=0, (n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))), ", ")) \\ _G. C. Greubel_ and _Vaclav Kotesovec_, Jul 28 2018
%Y A078738 B_{1, 1} = A000110, B_{2, 1} = A000262, B_{3, 1} = A020556 and B_{3, 3} = A069223. Row sums of A078740.
%Y A078738 Alternating row sums A090437.
%K A078738 nonn,easy
%O A078738 1,2
%A A078738 _N. J. A. Sloane_, Dec 21 2002
%E A078738 Edited by _Wolfdieter Lang_, Dec 23 2003