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
Offset: 0
Examples
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, 17, 33, 49, 65, 81, 97, ... 1, 161, 321, 481, 641, 801, 961, ... 1, 1351, 2841, 4471, 6241, 8151, 10201, ...
Links
- Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(3*j+1)^(n-2*j-1)/(j!*(n-3*j)!));
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.