Let a and c be integers. "Shew that" F := 1, F := a c - 1, F := 0 1 n 2 (2 n + a - 3) ((2 c - 1) (2 n + a - 4) (2 n + a - 2) + (a - 2) ) F n - 1 (----------------------------------------------------------------------- 2 - (n - 1) (n + a - 3) (2 n + a - 2) F )/(n (n + a - 2) (2 n + a - 4)) n - 2 (note cubic denominator) is an integer sequence. Distantly related: it appears that [ n + 4 n + 5 ] [ - n, -----, ----- | ] F [ 2 2 | -4] = 1,6,57,701,10147,164317,2888282,54047434, 3 2 [ | ] [ 3, 2 ] is an integer sequence. The smallest prime factors go {},2,3,701,73,37,2,2,7,2,3,3,5,2,23,37,65306671610636210891,2, 31,3,3,19,2,2,3,2,3,7,3,3,2,2,7,2,67,5,... . Likewise, [ n + 5 n + 6 ] [ - n, -----, ----- | ] 4 F [ 2 2 | -4] = 4,25,228,2620,35164,527663,8613004, 3 2 [ | ] [ 4, 2 ] [ n + 6 n + 7 ] [ - n, -----, ----- | ] 15 F [ 2 2 | -4] = 15,99,891,9825,125085,1772775,27303603, 3 2 [ | ] [ 5, 2 ] [ n + 7 n + 8 ] [ - n, -----, ----- | ] 5 F [ 2 2 | -4] = 5,29,230,2260,25921,334105,4717653, 3 2 [ | ] [ 5, 3 ] More generally, [ n + k + 1 n + k + 2 ] [ n + k + 2 n + k + 3 ] [-n,---------,---------| ] [-n,---------,---------| ] F [ 2 2 |-4] & F [ 2 2 |-4] 3 2[ | ] 3 2[ | ] [ k, 2 ] [ k, 3 ] appear to have bounded denominators. --rwg Does OEIS list sequences not proven integral?