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A384653
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384649.
4
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 9, 0, 1, 4, 9, 22, 56, 0, 1, 5, 14, 40, 134, 432, 0, 1, 6, 20, 64, 240, 1012, 3935, 0, 1, 7, 27, 95, 381, 1779, 9039, 40820, 0, 1, 8, 35, 134, 565, 2780, 15596, 92246, 471633, 0, 1, 9, 44, 182, 801, 4071, 23950, 156597, 1051558, 5980210, 0
OFFSET
0,8
FORMULA
A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(4*n-3*j+k,j)/(4*n-3*j+k) * A(n-j,j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 9, 22, 40, 64, 95, 134, ...
0, 56, 134, 240, 381, 565, 801, ...
0, 432, 1012, 1779, 2780, 4071, 5718, ...
0, 3935, 9039, 15596, 23950, 34515, 47786, ...
PROG
(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));
CROSSREFS
Columns k=0..1 give A000007, A384649.
Sequence in context: A384651 A091063 A384652 * A384654 A392379 A246935
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jun 06 2025
STATUS
approved