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

1,2

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

Table of n, a(n) for n=1..18.

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.

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

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 *)

Crossrefs

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

Sequence in context: A047737 A302535 A086403 * A346983 A271841 A361526

Adjacent sequences: A049117 A049118 A049119 * A049121 A049122 A049123

Keyword

easy,nonn,changed

Author

Wolfdieter Lang

Status

approved