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

1,2

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

Table of n, a(n) for n=1..19.

Wolfdieter Lang, On generalizations of the 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, arXiv:quant-ph/0402027, 2004.

P. Blasiak, K. A. Penson and A. I. Solomon, Combinatorial coherent states via normal ordering of bosons, arXiv:quant-ph/0311033, 2003.

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

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 *)

Crossrefs

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

Sequence in context: A047735 A096364 A210661 * A367423 A332236 A305981

Adjacent sequences: A049116 A049117 A049118 * A049120 A049121 A049122

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved