A306405 T(n,k) = k*Sum_{i=0..(n-k)/2} C(k,2*k+2*i-n)*C(k+2*i-1,i)/(k+i), triangle read by rows for n >= 1 and 1 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 4, 6, 4, 1, 2, 7, 10, 10, 5, 1, 5, 10, 18, 20, 15, 6, 1, 5, 19, 30, 39, 35, 21, 7, 1, 14, 28, 55, 72, 75, 56, 28, 8, 1, 14, 56, 93, 136, 151, 132, 84, 36, 9, 1, 42, 84, 174, 248, 300, 288, 217, 120, 45, 10, 1

Offset: 1

Author

Vladimir Kruchinin, Feb 13 2019

Keywords

• nonn
• tabl

Examples

			    1;
    1,  1;
    1,  2,  1;
    1,  3,  3,  1;
    2,  4,  6,  4,  1;
    2,  7, 10, 10,  5, 1;
    5, 10, 18, 20, 15, 6, 1;
		

Crossrefs

Cf. A000108, A274115.

Programs

Maple

# Seen as a (0,0)-based triangle:
gf := (2*(x + 1))/(sqrt(1 - 4*x^2) - 2*x*(x + 1)*y + 1):
serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
row := n -> seq(coeff(sery(n), y, j), j=0..n):
seq(lprint(row(n)), n=0..9); # Peter Luschny, Feb 14 2019

Maxima

T(n,k):=k*sum((binomial(k,2*k+2*i-n)*binomial(k+2*i-1,i))/(k+i),i,0,(n-k)/2);

Formula

G.f.: 1/(1-y*(x*(1+x)*(1-sqrt(1-4*x^2))/(2*x^2)))-1.