A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 20, 8, 0, 1, 5, 20, 54, 68, 16, 0, 1, 6, 30, 112, 252, 232, 32, 0, 1, 7, 42, 200, 656, 1188, 792, 64, 0, 1, 8, 56, 324, 1400, 3904, 5616, 2704, 128, 0, 1, 9, 72, 490, 2628, 10000, 23360, 26568, 9232, 256, 0

Offset

0,8

Links

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

Formula

T(0,k) = 1, T(1,k) = k; 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.

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

E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).

Example

Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  1,   2,    3,    4,     5, ...
  0,  2,   6,   12,   20,    30, ...
  0,  4,  20,   54,  112,   200, ...
  0,  8,  68,  252,  656,  1400, ...
  0, 16, 232, 1188, 3904, 10000, ...

Prog

(PARI) T(n, k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));
(PARI) T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;

Crossrefs

Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.

Main diagonal gives A084062.

Cf. A084061, A084097.

Sequence in context: A198793 A085388 A351339 * A294498 A292860 A265609

Adjacent sequences: A361429 A361430 A361431 * A361433 A361434 A361435

Keyword

nonn,tabl,easy

Author

Seiichi Manyama, Mar 11 2023

Status

approved