A094072 Coefficients arising in combinatorial field theory.

1, 6, 50, 615, 10192, 214571, 5544394, 171367020, 6208928376, 259542887975, 12356823485580, 662921411131909, 39714830070598204, 2636484537372437498, 192653800829700013970, 15405383160836582657251

Offset: 0

Author

N. J. A. Sloane, May 01 2004

Keywords

• nonn

References

• P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).

Links

• P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"

Crossrefs

Cf. A000085, A005425, A094070, A094071, A094073, A094074.

Cf. A000110.

Programs

Maple

with(combinat): seq(bell(n+1)*sum(k^(n+1-k)*binomial(n+1,k),k=1..n+1),n=0..18);

Mathematica

Table[BellB[n+1]Sum[Binomial[n+1,k]k^(n+1-k),{k,n+1}],{n,0,20}] (* Harvey P. Dale, Feb 05 2015 *)

Formula

a(n) = B(n+1)*Sum_{k=1..n+1} binomial(n+1, k)*k^(n+1-k), where B(n) are the Bell numbers (A000110). - Emeric Deutsch, Nov 23 2004

E.g.f.: exp(-1)*Sum_{k>=0} exp(k*x*exp(k*x))/k!. - Vladeta Jovovic, Sep 26 2006

Extensions

More terms from Emeric Deutsch, Nov 23 2004