OFFSET
1,2
LINKS
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
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, arXiv:quant-ph/0402027, 2002.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
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, 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, 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
MATHEMATICA
a[n_, k_] := 2^(n+k)*n!/(4^n*n*k!)*Sum[(j+k)*2^(j)*Binomial[j+k-1, k-1]*Binomial[2*n-j-k-1, n-1], {j, 0, n-k}]; a[n_] := Sum[a[n, k], {k, 1, n}]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jul 05 2013, after Emanuele Munarini *)
Table[Sum[BellY[n, k, (2 Range[n] - 1)!!], {k, n}], {n, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
PROG
(Maxima) a(n, k):=2^(n+k)*n!/(4^n*n*k!)*sum((j+k)*2^(j)*binomial(j+k-1, k-1)*binomial(2*n-j-k-1, n-1), j, 0, n-k); makelist(sum(a(n, k), k, 0, n), n, 1, 12); /* Emanuele Munarini, Jun 01 2012 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved