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 A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k). 5
 1, 0, 3, 1, 5, 10, 0, 10, 35, 35, 1, 14, 91, 189, 126, 0, 21, 189, 651, 924, 462, 1, 27, 351, 1749, 4026, 4290, 1716, 0, 36, 594, 4026, 13299, 22737, 19305, 6435, 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310, 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378, 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The scanned pages of Ser are essentially illegible, and the book is out of print and hard to locate. For Table IV on page 93, it is simplest to ignore the minus signs. The present triangle then matches all the given terms in that triangle, so it seems best to define the triangle by the recurrences given here, and to conjecture (strongly) that this is the same as Ser's triangle. REFERENCES J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. LINKS J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) EXAMPLE Triangle begins: 1, 0, 3, 1, 5, 10, 0, 10, 35, 35, 1, 14, 91, 189, 126, 0, 21, 189, 651, 924, 462, 1, 27, 351, 1749, 4026, 4290, 1716, 0, 36, 594, 4026, 13299, 22737, 19305, 6435, 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310, 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378, 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716, 0, 78, 2925, 47125, 403325, 2047123, 6529445, 13424203, 17753372, 14578928, 6760390, 1352078, 1, 90, 4005, 76075, 774775, 4738733, 18540523, 47971637, 82974178, 94853472,, 68747966, 28601650, 5200300, ... MAPLE SS := (n, k)->binomial(n, k)*binomial(n+k+1, k); T4:=proc(n, k) local i; global SS; option remember; if k=0 then return((1+(-1)^n)/2); fi; if n=0 then 0 else SS(n, k)-T4(n-1, k); fi; end; rho:=n->[seq(T4(n, k), k=0..n)]; for n from 0 to 14 do lprint(rho(n)); od: CROSSREFS Columns 1 and 2 are A176222 and A331429; the last three diagonals are A002739, A002737, A001700. Taking the component-wise sums of the rows by pairs give the triangle in A178303. Ser's tables I and III are A331430 and A331431 (both are still mysterious). Sequence in context: A055199 A146916 A146255 * A122366 A228781 A103327 Adjacent sequences:  A331429 A331430 A331431 * A331433 A331434 A331435 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Jan 17 2020 STATUS approved

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Last modified February 23 07:31 EST 2020. Contains 332159 sequences. (Running on oeis4.)