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A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

%I A182933 #13 May 12 2016 15:19:46
%S A182933 1,1,1,1,1,1,1,2,3,1,1,5,27,13,1,1,15,409,778,73,1,1,52,9089,104149,
%T A182933 37553,501,1,1,203,272947,25053583,57184313,2688546,4051,1,1,877,
%U A182933 10515147,9566642254,192052025697,56410245661,265141267,37633,1
%N A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.
%C A182933 These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
%C A182933 The square array starts for n>=0, k>=0:
%C A182933 n\k=0,1,..  A000012,A000262,A182934,...
%C A182933 0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%C A182933 1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
%C A182933 2: A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
%C A182933 3: A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
%C A182933 4:        : 1, 73, 37553, 57184313, 192052025697, ...
%C A182933 5:        : 1, 501, 2688546, 56410245661, ...
%C A182933 6:  ....  : 1, 4051, 265141267, 89501806774945, ...
%F A182933 Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).
%F A182933 Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).
%p A182933 A182933_AsSquareArray := proc(n,k) local r,s,i;
%p A182933 r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
%p A182933 exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
%p A182933 seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);
%t A182933 a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jul 29 2013 *)
%Y A182933 Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.
%K A182933 nonn,tabl
%O A182933 0,8
%A A182933 _Peter Luschny_, Mar 29 2011