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A362490

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 * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 4, 33, 161, 1, 1, 1, 1, 5, 49, 321, 1351, 1, 1, 1, 1, 6, 65, 481, 2841, 12391, 1, 1, 1, 1, 7, 81, 641, 4471, 31641, 153385, 1, 1, 1, 1, 8, 97, 801, 6241, 57751, 498849, 2388905, 1

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OFFSET

0,14

LINKS

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)

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)^3).

A_k(x) = exp(x - LambertW(-k*x^3/2 * exp(3*x))/3).

A_k(x) = ( -2 * LambertW(-k*x^3/2 * exp(3*x))/(k*x^3) )^(1/3) for k > 0.

EXAMPLE

Square array begins: