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

0,9

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,    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, ...

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, A143740, A125500, A362380.

Cf. A362378, A362394.

Sequence in context: A368119 A257493 A296526 * A259844 A112707 A196017

Adjacent sequences: A362374 A362375 A362376 * A362378 A362379 A362380

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 20 2023

Status

approved