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

0,7

Links

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.

Diagonals: A000012, A000217, A001477, A166830.

Sequence in context: A355624 A065369 A336404 * A077870 A294408 A358605

Adjacent sequences: A167769 A167770 A167771 * A167773 A167774 A167775

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 11 2009, corrected Nov 12 2009

Status

approved