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A120257 Triangle of Hankel transforms of certain binomial sums. 1
1, 2, -1, 3, -6, -1, 4, -20, -20, 1, 5, -50, -175, 70, 1, 6, -105, -980, 1764, 252, -1, 7, -196, -4116, 24696, 19404, -924, -1, 8, -336, -14112, 232848, 731808, -226512, -3432, 1, 9, -540, -41580, 1646568, 16818516, -24293412, -2760615, 12870, 1, 10, -825, -108900, 9343620, 267227532, -1447482465 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row k is the Hankel transform of Sum_{j=0..n} binomial(k+j, j). Absolute value is reversal of A103905. Diagonal and subdiagonals are essentially signed versions of the central coefficients of certain generalized Pascal-Narayana triangles (A007318, A001263, A056939, A056940, A056941).
LINKS
FORMULA
T(n, k) = (cos(Pi*k/2) - sin(Pi*k/2)) * Product_{j=0..n-k-1} C(2k+2+j, k+1)/C(k+1+j, j).
EXAMPLE
Triangle begins
1;
2, -1;
3, -6, -1;
4, -20, -20, 1;
5, -50, -175, 70, 1;
6, -105, -980, 1764, 252, -1;
7, -196, -4116, 24696, 19404, -924, -1;
8, -336, -14112, 232848, 731808, -226512, -3432, 1;
PROG
(PARI) T(n, k) = (-1)^((k+1)\2) * prod(j=0, n-k-1, binomial(2*k+2+j, k+1)/binomial(k+1+j, j)); \\ Michel Marcus, Jan 13 2022
CROSSREFS
Cf. A120258.
Sequence in context: A337414 A337410 A337389 * A337412 A337408 A059298
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Jun 13 2006
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)