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

0,14

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

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)).

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.

Example

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, ...

Prog

(PARI) T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));

Crossrefs

Columns k=0..3 give A000012, A362395, A362396, A362397.

Cf. A362277, A362377.

Sequence in context: A263152 A075463 A026518 * A345949 A379222 A348505

Adjacent sequences: A362391 A362392 A362393 * A362395 A362396 A362397

Keyword

sign,tabl

Author

Seiichi Manyama, Apr 20 2023

Status

approved