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A362078
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.
5
1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 22, 35, 1, 1, 9, 37, 105, 126, 1, 1, 11, 55, 215, 511, 462, 1, 1, 13, 76, 369, 1271, 2534, 1716, 1, 1, 15, 100, 571, 2526, 7651, 12720, 6435, 1, 1, 17, 127, 825, 4401, 17577, 46614, 64449, 24310, 1, 1, 19, 157, 1135, 7026, 34412, 123810, 286599, 328900, 92378
OFFSET
0,6
FORMULA
T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-n,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(n+j-1,j) * binomial(k*j,n-j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
3, 5, 7, 9, 11, 13, ...
10, 22, 37, 55, 76, 100, ...
35, 105, 215, 369, 571, 825, ...
126, 511, 1271, 2526, 4401, 7026, ...
PROG
(PARI) T(n, k) = sum(j=0, n, binomial(n+j-1, j)*binomial(k*j, n-j));
CROSSREFS
Columns k=0..3 give A088218, A213684, A362087, A362088.
Main diagonal gives A362080.
Cf. A362079.
Sequence in context: A336858 A086385 A295222 * A123162 A213998 A340970
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Apr 08 2023
STATUS
approved