A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 29, 36, 1, 1, 9, 58, 185, 137, 1, 1, 11, 97, 532, 1257, 543, 1, 1, 13, 146, 1161, 5209, 8925, 2219, 1, 1, 15, 205, 2156, 14849, 53347, 65445, 9285, 1, 1, 17, 274, 3601, 34041, 198729, 564499, 491825, 39587, 1

Offset: 0

Author

Seiichi Manyama, Aug 20 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,    1,     1,      1,       1,       1,        1, ...
  1,    3,     5,      7,       9,      11,       13, ...
  1,   10,    29,     58,      97,     146,      205, ...
  1,   36,   185,    532,    1161,    2156,     3601, ...
  1,  137,  1257,   5209,   14849,   34041,    67657, ...
  1,  543,  8925,  53347,  198729,  562551,  1330693, ...
  1, 2219, 65445, 564499, 2748641, 9608811, 27053749, ...
		

Crossrefs

Columns k=0..4 give A000012, A002212(n+1), A127846(n+1), A386362, A386389.

Main diagonal gives A386432.

Cf. A000108, A340968.

Programs

PARI

a(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*(j+1))!/((j+1)!*(j+2)!));

Formula

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

G.f. of column k: 2/(1 - (2*k+1)*x + sqrt((1-x) * (1-(4*k+1)*x))).

A(n,k) = (A340968(n+1,k) - A340968(n,k))/k for k > 0.

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

A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * (2*k+1)^(n-2*j) * binomial(n,2*j) * Catalan(j).