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

2,2

Links

Table of n, a(n) for n=2..56.

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

Example

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,
...

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 *)

Crossrefs

Edges of triangle are A006882 and A129890.

Sequence in context: A349967 A321985 A301524 * A253790 A372430 A053351

Adjacent sequences: A202209 A202210 A202211 * A202213 A202214 A202215

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 14 2011

Status

approved