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A351703
Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).
6
1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730
OFFSET
0,6
FORMULA
T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * T(j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, ...
4, 1, 0, 0, 0, 0, ...
21, 3, 1, 0, 0, 0, ...
148, 12, 4, 1, 0, 0, ...
1305, 70, 10, 5, 1, 0, ...
13806, 465, 40, 15, 6, 1, ...
PROG
(PARI) T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
(PARI) T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022
CROSSREFS
Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.
Sequence in context: A058710 A281891 A124539 * A369923 A249094 A096501
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 20 2022
STATUS
approved