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A072678 Generalized Bell numbers B_{4,2}. 2
1, 21, 1045, 93289, 12975561, 2581284541, 693347907421, 241253367679185, 105394372192969489, 56410454014314490981, 36271084122927079387941, 27567930377271475039277881, 24435533594428382909107147225 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
FORMULA
a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
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.
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
a(n) = A052852(2*n-1). - Mark van Hoeij, Sep 05 2022
MAPLE
f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
map(f, [$1..30]); # Robert Israel, May 23 2016
MATHEMATICA
a[n_] := n*(2n-1)!*Hypergeometric1F1[2-2n, 3, -1]; Array[a, 30] (* Jean-François Alcover, Sep 01 2016 *)
CROSSREFS
Cf. A090439 (alternating row sums of A090438).
Sequence in context: A186392 A360505 A263994 * A218821 A012153 A231521
KEYWORD
nonn
AUTHOR
Karol A. Penson, Jul 01 2002
EXTENSIONS
Edited by Wolfdieter Lang, Dec 23 2003
STATUS
approved

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Last modified February 22 13:38 EST 2024. Contains 370256 sequences. (Running on oeis4.)