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Row sums of triangle A035342 and array A134144.
6

%I #29 Nov 10 2016 03:48:50

%S 1,4,25,211,2236,28471,422899,7173580,136750051,2893057381,

%T 67241818876,1702829138209,46659181547785,1375237342827076,

%U 43380198327693361,1458027134026128691,52014149849253158284,1962794208713975883415

%N Row sums of triangle A035342 and array A134144.

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.

%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2002.

%H P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering via substitutions and Sheffer-type polynomials</a>, arXiv:quant-ph/0501155, 2005.

%F E.g.f. exp(-1+1/sqrt(1-2*x))-1.

%F 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

%F 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

%F 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

%t 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_ *)

%t Table[Sum[BellY[n, k, (2 Range[n] - 1)!!], {k, n}], {n, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)

%o (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 */

%Y Cf. A000262, A049119, A049120.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_