A049119 Row sums of triangle A035469.

1, 5, 41, 465, 6721, 117941, 2433145, 57673281, 1543866945, 46052954821, 1514472783561, 54426342354385, 2121878761891201, 89187219264121525, 4020175011403931801, 193438800635132796161, 9895634072548245693825, 536284759396849853348101, 30691678336547328623916905

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Generalized Bell numbers B(4,1;n).

References

• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

Links

• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
• P. Blasiak, K. A. Penson and A. I. Solomon, Combinatorial coherent states via normal ordering of bosons.

Crossrefs

Cf. Generalized Bell numbers B(m, 1, n): A049118 (m=3), this sequence (m=4), A049120 (m=5), A049412 (m=6).

Programs

Mathematica

Drop[CoefficientList[Series[Exp[-1+1/(1-3*x)^(1/3)]-1,{x,0,19}],x]Range[0,19]!,1] (* Stefano Spezia, Mar 31 2025 *)

Formula

E.g.f.: exp(-1+1/(1-3*x)^(1/3))-1.

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A000110, A000262, A049118 and A049120. - Peter Bala, Nov 25 2011

a(n) = (1/e) * (-3)^n * n! * Sum_{k>=0} binomial(-k/3,n)/k!. - Seiichi Manyama, Jan 17 2025