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A361830
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).
4
1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620
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} 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, ...
2, 2, 2, 2, 2, 2, ...
6, 8, 10, 12, 14, 16, ...
20, 32, 46, 62, 80, 100, ...
70, 136, 226, 342, 486, 660, ...
252, 592, 1136, 1932, 3030, 4482, ...
PROG
(PARI) T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));
CROSSREFS
Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Sequence in context: A357124 A210227 A208757 * A133643 A008305 A208763
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Mar 26 2023
STATUS
approved