A049120 Row sums of triangle A049029.

1, 6, 61, 871, 15996, 358891, 9509641, 290528316, 10051973371, 388433817091, 16579346005806, 774580047063901, 39313104018590221, 2153825039102763846, 126681355435102649161, 7961385691338995966371, 532402860878855993673036, 37746950872336992298209151

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Generalized Bell numbers B(5,1;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.

Crossrefs

Cf. Generalized Bell numbers B(m, 1, n): A049118 (m=3), A049119 (m=4), this sequence (m=5), A049412 (m=6).

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Exp[-1+1/Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2019 *)

Formula

E.g.f. exp(-1+1/(1-4*x)^(1/4))-1.

Representation of a(n) as the n-th moment of a positive function on positive half-axis (Stieltjes moment problem), in Maple notation: a(n)=int(x^n*exp(-1)*exp(-1/4*x)*(1/96*x*hypergeom([],[5/4, 3/2, 7/4, 2],1/1024*x)+ 1/8*4^(3/4)*x^(1/4)/Pi*2^(1/2)*GAMMA(3/4)*hypergeom([],[1/4, 1/2,3/4, 5/4],1/1024*x)+1/8*4^(1/2)*x^(1/2)/Pi^(1/2)*hypergeom([],[1/2, 3/4, 5/4,3/2],1/1024*x)+1/24*4^(1/4)*x^(3/4)/GAMMA(3/4)*hypergeom([],[3/4, 5/4, 3/2,7/4],1/1024*x))/x, x=0..infinity),n=1,2... . - Karol A. Penson, Dec 16 2007

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A000110, A000262, A049118 and A049119. - Peter Bala, Nov 25 2011

a(n) = (1/e) * (-4)^n * n! * Sum_{k>=0} binomial(-k/4,n)/k!. - Seiichi Manyama, Jan 17 2025