A362483 - OEIS

A362483

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2*j+1)^(n-j-1) / (j! * (n-2*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 10, 1, 1, 1, 4, 19, 70, 1, 1, 1, 5, 28, 169, 646, 1, 1, 1, 6, 37, 298, 2041, 7576, 1, 1, 1, 7, 46, 457, 4186, 30811, 106744, 1, 1, 1, 8, 55, 646, 7081, 74116, 560827, 1761628, 1, 1, 1, 9, 64, 865, 10726, 141901, 1578340, 11957905, 33361948, 1

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OFFSET

0,9

LINKS

Table of n, a(n) for n=0..65.

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^2/2 * A_k(x)^2).

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

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

EXAMPLE

Square array begins: