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A361834
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).
4
1, 1, 2, 1, 2, 6, 1, 2, 4, 20, 1, 2, 2, 8, 70, 1, 2, 0, -2, 16, 252, 1, 2, -2, -10, -14, 32, 924, 1, 2, -4, -16, -22, -32, 64, 3432, 1, 2, -6, -20, -10, 12, -30, 128, 12870, 1, 2, -8, -22, 20, 118, 174, 64, 256, 48620, 1, 2, -10, -22, 66, 242, 304, 344, 346, 512, 184756
OFFSET
0,3
FORMULA
G.f. of column k: 1/sqrt(1 - 4*x*(1-x)^k).
n*T(n,k) = 2 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, ...
6, 4, 2, 0, -2, -4, -6, ...
20, 8, -2, -10, -16, -20, -22, ...
70, 16, -14, -22, -10, 20, 66, ...
252, 32, -32, 12, 118, 242, 342, ...
924, 64, -30, 174, 304, 82, -678, ...
PROG
(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(2*j, j)*binomial(k*j, n-j));
CROSSREFS
Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.
Main diagonal gives A361835.
Cf. A361830.
Sequence in context: A083773 A129116 A096179 * A166350 A357124 A210227
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Mar 26 2023
STATUS
approved