A202212 Triangle read by rows: T(n,k) (1 <= k <= n-1, n >= 2) = d(2*(n-k)-1)*(d(2*n-2)/d(2*(n-k)-2) - d(2*n-3)/d(2*(n-k)-3)), where d = A006882 is the double factorial function.

1, 3, 5, 15, 27, 33, 105, 195, 261, 279, 945, 1785, 2475, 2925, 2895, 10395, 19845, 28035, 34425, 37935, 35685, 135135, 259875, 371385, 465255, 533925, 562275, 509985, 2027025, 3918915, 5644485, 7158375, 8390025, 9218475, 9401805, 8294895, 34459425, 66891825, 96891795, 123898005, 147093975, 165209625, 176067675, 175313565, 151335135, 654729075, 1274998725, 1854727875, 2385808425, 2857013775, 3252014325, 3545408475, 3693650625, 3609649575, 3061162125

Offset: 2

Author

N. J. A. Sloane, Dec 14 2011

Keywords

• nonn
• tabl

Examples

			Triangle begins
1,
3, 5,
15, 27, 33,
105, 195, 261, 279,
945, 1785, 2475, 2925, 2895,
10395, 19845, 28035, 34425, 37935, 35685,
135135, 259875, 371385, 465255, 533925, 562275, 509985,
...
		

Links

• N. Ochiumi, On the total sum of number of nodes covering a given number of leaves in an unordered binary tree.

Crossrefs

Edges of triangle are A006882 and A129890.

Programs

Maple

d:=doublefactorial;
a:=(n,k)-> d(2*(n-k)-1)*(d(2*n-2)/d(2*(n-k)-2) - d(2*n-3)/d(2*(n-k)-3));
f:=n->[seq(a(n,k),k=1..n-1)];
for n from 1 to 10 do lprint(f(n)); od:

Mathematica

a[n_, k_] := (2*(n-k)-1)!!*((2*n-2)!!/(2*(n-k)-2)!!-(2*n-3)!!/(2*(n-k)-3)!!); Table[a[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)