A362490 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 4, 33, 161, 1, 1, 1, 1, 5, 49, 321, 1351, 1, 1, 1, 1, 6, 65, 481, 2841, 12391, 1, 1, 1, 1, 7, 81, 641, 4471, 31641, 153385, 1, 1, 1, 1, 8, 97, 801, 6241, 57751, 498849, 2388905, 1

Offset

0,14

Links

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)

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^3/6 * A_k(x)^3).

A_k(x) = exp(x - LambertW(-k*x^3/2 * exp(3*x))/3).

A_k(x) = ( -2 * LambertW(-k*x^3/2 * exp(3*x))/(k*x^3) )^(1/3) for k > 0.

Example

Square array begins:
  1,    1,    1,    1,    1,    1,     1, ...
  1,    1,    1,    1,    1,    1,     1, ...
  1,    1,    1,    1,    1,    1,     1, ...
  1,    2,    3,    4,    5,    6,     7, ...
  1,   17,   33,   49,   65,   81,    97, ...
  1,  161,  321,  481,  641,  801,   961, ...
  1, 1351, 2841, 4471, 6241, 8151, 10201, ...

Prog

(PARI) T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(3*j+1)^(n-2*j-1)/(j!*(n-3*j)!));

Crossrefs

Columns k=0..3 give A000012, A362477, A362478, A362479.

Cf. A362378.

Sequence in context: A062277 A362378 A204929 * A118210 A061399 A161856

Adjacent sequences: A362487 A362488 A362489 * A362491 A362492 A362493

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 22 2023

Status

approved