A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.
1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 6, 8, 6, 3, 1, 18, 24, 18, 10, 4, 1, 57, 75, 57, 33, 15, 5, 1, 186, 243, 186, 111, 54, 21, 6, 1, 622, 808, 622, 379, 193, 82, 28, 7, 1, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 7338, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1
Offset: 0
Examples
Triangle begins:
1;
0, 1;
1, 1, 1;
2, 3, 2, 1;
6, 8, 6, 3, 1;
18, 24, 18, 10, 4, 1;
57, 75, 57, 33, 15, 5, 1;
186, 243, 186, 111, 54, 21, 6, 1;
622, 808, 622, 379, 193, 82, 28, 7, 1;
2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1;
Production matrix begins:
0, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
... - _Philippe Deléham_, Mar 03 2013
Links
- Reinhard Zumkeller, Rows n=0..125 of triangle, flattened
Crossrefs
Programs
-
Haskell
import Data.List (genericIndex) a167772 n k = genericIndex (a167772_row n) k a167772_row n = genericIndex a167772_tabl n a167772_tabl = [1] : [0, 1] : map (\xs@(:x:) -> x : xs) (tail a065602_tabl) -- Reinhard Zumkeller, May 15 2014 -
Mathematica
A065602[n_, k_]:= A065602[n,k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n-j) -k-1), {j, 0, (n-k)/2}]; T[n_, k_]:= If[k==0, A065602[n+1,3] + Boole[n==0], A065602[n+1, k+1]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2022 *)
-
SageMath
def A065602(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) ) def A167772(n,k): if (k==0): return A065602(n+1,3) + bool(n==0) else: return A065602(n+1,k+1) flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022
Formula
Sum_{k=0..n} T(n, k) = A000958(n+1).
From Philippe Deléham, Nov 12 2009: (Start)
Sum_{k=0..n} T(n,k)*2^k = A014300(n).
Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)
For n > 0: T(n,0) = A065602(n+1,3), T(n,k) = A065602(n+1,k+1), k = 1..n. - Reinhard Zumkeller, May 15 2014