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A362378
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).
7
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 9, 1, 1, 1, 1, 4, 17, 41, 1, 1, 1, 1, 5, 25, 81, 191, 1, 1, 1, 1, 6, 33, 121, 441, 1191, 1, 1, 1, 1, 7, 41, 161, 751, 3641, 9353, 1, 1, 1, 1, 8, 49, 201, 1121, 7351, 33825, 77897, 1
OFFSET
0,14
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)).
A_k(x) = exp(x - LambertW(-k*x^3/6 * exp(x))).
A_k(x) = -6 * LambertW(-k*x^3/6 * exp(x))/(k*x^3) for k > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 9, 17, 25, 33, 41, 49, ...
1, 41, 81, 121, 161, 201, 241, ...
1, 191, 441, 751, 1121, 1551, 2041, ...
PROG
(PARI) T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(j+1)^(n-2*j-1)/(j!*(n-3*j)!));
CROSSREFS
Columns k=0..3 give A000012, A362381, A362390, A362391.
Sequence in context: A362043 A337220 A062277 * A204929 A362490 A118210
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Apr 20 2023
STATUS
approved