|
| |
| |
|
|
|
1, 4, 25, 211, 2236, 28471, 422899, 7173580, 136750051, 2893057381, 67241818876, 1702829138209, 46659181547785, 1375237342827076, 43380198327693361, 1458027134026128691, 52014149849253158284, 1962794208713975883415
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
LINKS
| P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
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, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials
|
|
|
FORMULA
| E.g.f. exp(-1+1/sqrt(1-2*x))-1.
Representation of a(n) as n-th moment of a positive function on (0, infinity): a(n)=int(x^n* (x/2)^(-1/2)*exp(-x/2)*(2*hypergeom([], [3/2, 1/2], 1/8*x)/Pi^(1/2)+1/2*sqrt(2)*sqrt(x)*hypergeom([], [2, 3/2], 1/8*x))/(4*exp(1)), x=0..infinity), n=1, 2... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Jun 27, 2002
Asymptotic expansion for large n: a(n) -> 2^(1/6)*(n^(-1/3) + 2^(-7/3)*n^(-2/3) + O(1/n))*(2*n)^n*exp(-n+(3/2)*(2*n)^(1/3))/(sqrt(3)*exp(1)); (the nature of this approximation of a(n) is the same as that of Stirling approximation of n! ). - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 02, 2002
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^3*d/dx. Cf. A000110, A000262, A049119 and A049120 - Peter Bala, Nov 25 2011
|
|
|
CROSSREFS
| Cf. A000262. A049119, A049120.
Sequence in context: A203219 A064299 A038174 * A047733 A198198 A007830
Adjacent sequences: A049115 A049116 A049117 * A049119 A049120 A049121
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
|
| |
|
|
