A090440 -id:A090440 - cp's OEIS frontend

A091028 Alternating row sums of array A090440 (generalized Stirling2 array (4,3)).

1, -1, -299, 40751, 5067841, -7323445921, -999631061099, 8071239253153439, 3226932423143680609, -35637793578511857426241, -56681720635442893862661899, 428341845237867375193339807439, 2249822227541602344980346463546081

Offset: 1

Wolfdieter Lang, Dec 23 2003

Cf. A070531 (row sums of A090440).

a(n)=-sum(((-1)^k)*A090440(n, k), k=3..3*n), n>=1.

A070531 Generalized Bell numbers B_{4,3}.

Original entry on oeis.org

1, 73, 16333, 8030353, 7209986401, 10541813012041, 23227377813664333, 72925401604382826913, 312727862321385812968033, 1772004571987390827615327241, 12917715377912025572750844722221, 118521774439119390334062953438350513, 1343761301099219856651740487814621053313

Offset: 1

Karol A. Penson, May 02 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..200
P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A091028 (alternating row sums of A090440).

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, ] = 0; Table[Sum[a[n, k] , {k, 3, 3 n}], {n, 1, 9}] (* _Jean-François Alcover, Sep 01 2011 *)

In Maple notation, a(n) = (1/12)*n!*(n+1)!*(n+2)!*hypergeom([n+1, n+2, n+3], [2, 3, 4], 1)/exp(1).

a(n) = Sum_{k=3..3*n} A090440(n, k) = (Sum_{k>=3} (1/k!)*Product_{j=1..n} fallfac(k+(j-1)*(4-3), 3))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

Edited by Wolfdieter Lang, Dec 23 2003

cp's OEIS Frontend

A091028 Alternating row sums of array A090440 (generalized Stirling2 array (4,3)).

Views

Author

Keywords

Crossrefs

Formula

A070531 Generalized Bell numbers B_{4,3}.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions