A362377 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, 2, 1, 1, 1, 3, 7, 1, 1, 1, 4, 13, 34, 1, 1, 1, 5, 19, 85, 216, 1, 1, 1, 6, 25, 154, 701, 1696, 1, 1, 1, 7, 31, 241, 1456, 7261, 15898, 1, 1, 1, 8, 37, 346, 2481, 18136, 89125, 173468, 1, 1, 1, 9, 43, 469, 3776, 35761, 260002, 1277865, 2161036, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, ... 1, 7, 13, 19, 25, 31, 37, ... 1, 34, 85, 154, 241, 346, 469, ... 1, 216, 701, 1456, 2481, 3776, 5341, ... 1, 1696, 7261, 18136, 35761, 61576, 97021, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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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.