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

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, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset

0,8

Comments

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).

The square array starts for n>=0, k>=0:

n\k=0,1,.. A000012,A000262,A182934,...

0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...

2: A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...

3: A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...

4: : 1, 73, 37553, 57184313, 192052025697, ...

5: : 1, 501, 2688546, 56410245661, ...

6: .... : 1, 4051, 265141267, 89501806774945, ...

Links

Table of n, a(n) for n=0..44.

Formula

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).

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).

Maple

A182933_AsSquareArray := proc(n, k) local r, s, i;
r := [seq(n+1, i=1..k)]; s := [seq(1, i=1..k-1), 2];
exp(-x)*n!^k*hypergeom(r, s, x); round(evalf(subs(x=1, %), 99)) end:
seq(lprint(seq(A182933_AsSquareArray(n, k), k=0..6)), n=0..6);

Mathematica

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 *)

Crossrefs

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.

Sequence in context: A131739 A011151 A140878 * A342673 A365463 A068348

Adjacent sequences: A182930 A182931 A182932 * A182934 A182935 A182936

Keyword

nonn,tabl

Author

Peter Luschny, Mar 29 2011

Status

approved