A362394 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 * (j+1)^(n-j-1) / (j! * (n-2*j)!).
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -5, 1, 1, 1, -2, -11, -14, 1, 1, 1, -3, -17, -11, 56, 1, 1, 1, -4, -23, 10, 381, 736, 1, 1, 1, -5, -29, 49, 976, 2461, 1114, 1, 1, 1, -6, -35, 106, 1841, 3736, -21083, -45156, 1, 1, 1, -7, -41, 181, 2976, 3121, -106910, -449623, -428660, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... 1, 0, -1, -2, -3, -4, -5, ... 1, -5, -11, -17, -23, -29, -35, ... 1, -14, -11, 10, 49, 106, 181, ... 1, 56, 381, 976, 1841, 2976, 4381, ... 1, 736, 2461, 3736, 3121, -824, -9539, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
-
PARI
T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));
Formula
E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x - k*x^2/2 * A_k(x)).
A_k(x) = exp(x - LambertW(k*x^2/2 * exp(x))).
A_k(x) = 2 * LambertW(k*x^2/2 * exp(x))/(k*x^2) for k > 0.