On 8/4/07, N. J. A. Sloane <njas@research.att.com> wrote:
Fred, Thanks for the suggestion about A005493
Also A000296 ?
- i will add that formula.
I will also add a couple of other (B+k)^n sequences
See below.
You said the OEIS server was down - it seems OK to me - can you try again?
OEIS is OK at present. But for a couple of hours at least last night (say 0200 - 0400 hrs BST) I could get no response from it at all, although I had no problem reaching other USA sites. Heavy traffic at your end, maybe? Attached a recursive Maple function for these generalised Bell numbers [you could also use (B+k)^n, a Stirling number sum formula, or for speed a Dobinski-style series]; also a short table (which would be better transposed). The reference E. A. Enneking, J. C. Ahuja \sl Generalized Bell numbers \rm Fib. Quart., \bf 14 \rm 67-73 (1976) _may_ previously have discussed these sequences: I have not yet obtained sight of it, so the G_k(n) notation is necessarily provisional. Yours, Fred Lunnon -------------------------------------------------------------------------------- Table G_k(n) n\k_____-5____-4___-3__-2__-1___ 0____ 1____ 2____ 3______ 4______ 5 0 ______ 1____ 1___ 1__ 1__ 1___ 1____ 1____ 1____ 1______ 1______ 1 1 ______-4____-3___-2__-1__ 0___ 1____ 2____ 3____ 4______ 5______ 6 2 _____ 17___ 10___ 5__ 2__ 1___ 2____ 5___ 10___ 17_____ 26_____ 37 3 _____-75___-35__-13__-3__ 1___ 5___ 15___ 37___ 77____ 141____ 235 4 ____ 340__ 127__ 36__ 7__ 4__ 15___ 52__ 151__ 372____ 799___ 1540 5 ___-1573__-472_-101_-10_ 11__ 52__ 203__ 674_ 1915___ 4736__ 10427 6 ___ 7393_ 1787_ 293_ 31_ 41_ 203__ 877_ 3263 10481__ 29371__ 73013 7 __-35178_-6855_-848_-21 162_ 877_ 4140 17007 60814_ 190497_ 529032 8 _ 169035 26572 2523 204 715 4140 21147 94828 37293 1291020 3967195 -------------------------------------------------------------------------------- # Generalised Bell number G_k(n) via recursion G^{n+1} = (G+1)^n + k G^n bell_gen := proc (n, k) local i,j,beli,bel,beln; if n < 0 then beln := floor(1.0/0.0) # oo for n < 0 else bel := array(0..n); bel[0] := 1; for i from 0 to n-1 do beli := k*bel[i]; for j from 0 to i do beli := beli + bel[j]*binomial(i, j) od; bel[i+1] := beli od; beln := bel[n] fi; beln end;