A094074 Coefficients arising in combinatorial field theory.

1, 5, 129, 7485, 755265, 116338005, 25263540225, 7328358482445, 2730934406225025, 1269262202389906725, 718835160819268317825, 486853691847850902700125, 388278919916351519293663425

Offset: 0

Author

N. J. A. Sloane, May 01 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..225
• 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"
• 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).
• A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.

Crossrefs

Equals A001147(n) * A093620(n).

Cf. A000085, A005425, A094071, A094072, A094073.

Programs

Mathematica

a[n_] := (2n)! SeriesCoefficient[(1-x^2)^(-1/2) Exp[2x^2/(1-x^2)], {x, 0, 2n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 11 2018 *)

Formula

a(n) = (2n)!/(2^n*n!) * h(2n, 2), with h(n, x) the polynomials in A099174.

E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = (1-x^2)^(-1/2) * exp(2x^2/(1-x^2)).

Extensions

Edited and extended by Ralf Stephan, Oct 14 2004.