cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jan 31 2021

Keywords

Examples

			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, ...
		

Crossrefs

Columns k=0..4 give A000012, A007317(n+1), A162326(n+1), A337167, A386387.
Main diagonal gives A338979.

Programs

  • Maple
    T_row := n -> k -> hypergeom([1/2, -n], [2], -4*k): for n from 0 to 6 do seq(simplify(T_row(n)(k)), k = 0..6) od; # Peter Luschny, Aug 27 2025
  • 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 *)
    A340968[n_, k_] := Hypergeometric2F1[1/2, -n, 2, -4*k]; Table[A340968[n, k], {n, 0, 6}, {k, 0, 7}] (* row-wise *) (* Peter Luschny, Aug 27 2025 *)
  • 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));

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
T_row(n) = k -> hypergeom([1/2, -n], [2], -4*k). - Peter Luschny, Aug 27 2025