A362019 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * Sum_{j=0..n} (-k*j)^j * binomial(n,j).
1, 1, -1, 1, 0, 1, 1, 1, 3, -1, 1, 2, 13, 17, 1, 1, 3, 31, 173, 169, -1, 1, 4, 57, 629, 3321, 2079, 1, 1, 5, 91, 1547, 18025, 81529, 31261, -1, 1, 6, 133, 3089, 58993, 662639, 2443333, 554483, 1, 1, 7, 183, 5417, 147081, 2888979, 29752957, 86475493, 11336753, -1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... -1, 0, 1, 2, 3, 4, ... 1, 3, 13, 31, 57, 91, ... -1, 17, 173, 629, 1547, 3089, ... 1, 169, 3321, 18025, 58993, 147081, ... -1, 2079, 81529, 662639, 2888979, 8998399, ...
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Crossrefs
Programs
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PARI
T(n, k) = (-1)^n*sum(j=0, n, (-k*j)^j*binomial(n, j));
Formula
E.g.f. of column k: exp(-x) / (1 + LambertW(-k*x)).
G.f. of column k: Sum_{j>=0} (k*j*x)^j / (1 + x)^(j+1).