A362483 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 * (2*j+1)^(n-j-1) / (j! * (n-2*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 10, 1, 1, 1, 4, 19, 70, 1, 1, 1, 5, 28, 169, 646, 1, 1, 1, 6, 37, 298, 2041, 7576, 1, 1, 1, 7, 46, 457, 4186, 30811, 106744, 1, 1, 1, 8, 55, 646, 7081, 74116, 560827, 1761628, 1, 1, 1, 9, 64, 865, 10726, 141901, 1578340, 11957905, 33361948, 1

Offset

0,9

Links

Table of n, a(n) for n=0..65.

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)^2).

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

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

Example

Square array begins:
  1,   1,    1,    1,    1,     1, ...
  1,   1,    1,    1,    1,     1, ...
  1,   2,    3,    4,    5,     6, ...
  1,  10,   19,   28,   37,    46, ...
  1,  70,  169,  298,  457,   646, ...
  1, 646, 2041, 4186, 7081, 10726, ...

Prog

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

Crossrefs

Columns k=0..3 give A000012, A362474, A143768, A362475.

Cf. A362377, A362490.

Sequence in context: A240472 A366836 A007442 * A054772 A294616 A085384

Adjacent sequences: A362480 A362481 A362482 * A362484 A362485 A362486

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 21 2023

Status

approved