A086614 Triangle read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.

1, 2, 1, 3, 4, 2, 4, 10, 12, 5, 5, 20, 42, 40, 14, 6, 35, 112, 180, 140, 42, 7, 56, 252, 600, 770, 504, 132, 8, 84, 504, 1650, 3080, 3276, 1848, 429, 9, 120, 924, 3960, 10010, 15288, 13860, 6864, 1430, 10, 165, 1584, 8580, 28028, 57330, 73920, 58344, 25740

Offset: 0

Author

Paul D. Hanna, Jul 24 2003

Keywords

• nonn
• tabl
• easy

Examples

			Rows:
{1},
{2, 1},
{3, 4,    2},
{4, 10,  12,    5},
{5, 20,  42,   40,   14},
{6, 35, 112,  180,  140,   42},
{7, 56, 252,  600,  770,  504,  132},
{8, 84, 504, 1650, 3080, 3276, 1848, 429}, ...
		

Crossrefs

T(n,n) = A000108(n).

Cf. A086615 (antidiagonal sums), A086616 (row sums), A086617, A000292 (column 1), A277935 (column 2), A000580 (column 3 divided by 5), A000582 (column 4 divided by 14).

Programs

Maple

T := (n,k) -> `if`(k=0, n+1, binomial(2*k, k-1)*binomial(n+k+1, n-k)/k):
for n from 0 to 8 do seq(T(n,k), k=0..n) od; # Peter Luschny, Jan 26 2018

Formula

T(n,k) = binomial(2*k, k-1)*binomial(n+k+1, n-k) / k for k > 0. # Peter Luschny, Jan 26 2018