I've mentioned this generalisation of Bell numbers before in math-fun, so I'll try to avoid taking up too much bandwidth repeating myself. There is a fuller discussion of G_k(n), including a short table and Maple procedures for computing them, in Maple Advisor at http://mapleadvisor.com/cgi-bin/moin.cgi/Bell%20Numbers . They seem to have made their original appearance (with a different origin, for positive k only) in a paper by Gill Williamson (1976), where they were applied to "ranking" or indexing set partitions by natural numbers. I encountered them in a completely different setting: the efficient computation of Bell numbers B(n) modulo p, for small prime p and large natural n. Case k = 0 is simply the usual Bell numbers, and other small values of k also yield well-known sequences: G_{-2}(n) = 1,-1,2,-3,7,-10,31,-21,204,307,2811,... [OEIS A126617]; (*) G_{-1}(n) = 1,0,1,1,4,11,41,162,715,3425,... [OEIS A000296]; (*) G_0(n) = 1,1,2,5,15,52,203,877,... Bell numbers B(n) [OEIS A000110]; G_1(n) = 1,2,5,15,52,203,877,... shifted Bell numbers B(n+1); G_2(n) = 1,3,10,37,151,674,3263,... [OEIS A005493]; G_3(n) = 1,4,17,77,372,1915,10481,60814,... [OEIS A005494]; G_4(n) = 1,5,26,141,799,4736,29371,... [OEIS A045379]; the 2-D sequence is found at [OEIS ], although for k >= 0 only, and so omitting the special cases marked (*) above. A simple O(n^2) recursive rule for generating these numbers is G_k(n+1) = G_{k+1}(n) + k G_k(n) with G_k(0) = 1; or for somewhat larger n, the Dobinski summation is available: G_m (n) = (1/e) \sum_{i >= 0} (i+m)^n / i!; and there are various other nice umbral relations involving them. In this capacity, a generalisation of Touchard's recurrence has corollary: for all k and prime p, G_k(p) = k+2 (mod p); leading immediately to a strengthening of the (somewhat computationally impractical) converse criterion suggested a year or so ago by David Wilson. That possibility was dashed by the counter-example n = 21361 = 41 x 521, for which B(n) = 2 (mod n) --- case k = 0 of the above. However, taking k = 1 we find for this n, G_1(n) = B(n+1) = 3 (mod n) is actually false; so (tongue firmly in cheek) we proudly unveil the new improved Wilson-Lunnon primality-test (conjectural) --- Is there any composite number n for which simultaneously B(n) = 2 (mod n) and B(n+1) = 3 mod n ? And if so, is there any n for which G_k(n) = k+2 (mod n) for all 0 <= k < n ? [Note that, for given n, G_k(n) mod n qua function of k has period n.] A second excuse for my telling you all this concerns a striking property of the "new" sequences for k < 0: for small n their signs alternate with n, but from some n >= 2 m + 1 say, the terms apparently become all positive. For 0 < -k <= 40, these critical values are at m = 0,0,4,8,13,18,24,30,36,42, 49,56,63,70,77,85,92,100,108,115, 123,131,140,148,156,164,173,181,190,198, 207,216,225,234,243,252,261,270,279,288, 297,... Notice that m = m(-k) appears remarkably close to the ceiling of 2|k| log |k| + 3; and what I should like to know is --- are these things true; and if so, why? Fred Lunnon [12/11/08]