A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j*binomial(n,j)*Catalan(j).

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 15, 1, 1, 5, 25, 71, 51, 1, 1, 6, 41, 199, 441, 188, 1, 1, 7, 61, 429, 1795, 2955, 731, 1, 1, 8, 85, 791, 5073, 17422, 20805, 2950, 1, 1, 9, 113, 1315, 11571, 64469, 177463, 151695, 12235, 1

Offset

0,5

Links

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

Formula

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

A_k(x) = 2/( 1 - x + sqrt((1 - x) * (1 - (4*k+1)*x)) ).

T(n,k) = 1 + k * Sum_{j=0..n-1} T(j,k) * T(n-1-j,k).

(n+1) * T(n,k) = 2 * ((2*k+1) * n - k) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.

E.g.f. of column k: exp((2*k+1)*x) * (BesselI(0,2*k*x) - BesselI(1,2*k*x)). - Ilya Gutkovskiy, Feb 01 2021

Example

Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   2,    3,     4,     5,      6, ...
  1,   5,   13,    25,    41,     61, ...
  1,  15,   71,   199,   429,    791, ...
  1,  51,  441,  1795,  5073,  11571, ...
  1, 188, 2955, 17422, 64469, 181776, ...

Mathematica

T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * CatalanNumber[j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)

Prog

(PARI) {T(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*j)!/(j!*(j+1)!))}
(PARI) {T(n, k) = 1+k*sum(j=0, n-1, T(j, k)*T(n-1-j, k))}

Crossrefs

Columns k=0..3 give A000012, A007317(n+1), A162326(n+1), A337167.

Main diagonal gives A338979.

Cf. A000108, A290605, A340970.

Sequence in context: A112338 A111672 A323206 * A128198 A320031 A123349

Adjacent sequences: A340965 A340966 A340967 * A340969 A340970 A340971

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 31 2021

Status

approved