A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0

Offset

0,9

Links

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

Formula

T(0,k) = 0, T(1,k) = 1; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).

T(n,k) = ((k + sqrt(k))^n - (k - sqrt(k))^n)/(2 * sqrt(k)) for k > 0.

G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).

E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

Example

Square array begins:
  0,  0,   0,   0,    0,    0, ...
  1,  1,   1,   1,    1 ,   1, ...
  0,  2,   4,   6,    8,   10, ...
  0,  4,  14,  30,   52,   80, ...
  0,  8,  48, 144,  320,  600, ...
  0, 16, 164, 684, 1936, 4400, ...

Prog

(PARI) T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

Crossrefs

Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.

Main diagonal gives A360766.

Cf. A361432.

Sequence in context: A072458 A256282 A258256 * A285638 A325667 A067310

Adjacent sequences: A361287 A361288 A361289 * A361291 A361292 A361293

Keyword

nonn,easy,tabl

Author

Seiichi Manyama, Mar 11 2023

Status

approved