A155761 Riordan array (c(2*x^2), x*c(2*x^2)) where c(x) is the g.f. of A000108.

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 8, 0, 6, 0, 1, 0, 20, 0, 8, 0, 1, 40, 0, 36, 0, 10, 0, 1, 0, 112, 0, 56, 0, 12, 0, 1, 224, 0, 224, 0, 80, 0, 14, 0, 1, 0, 672, 0, 384, 0, 108, 0, 16, 0, 1, 1344, 0, 1440, 0, 600, 0, 140, 0, 18, 0, 1

Offset: 0

Author

Paul Barry, Jan 26 2009

Keywords

• easy
• nonn
• tabl

Comments

Inverse of Riordan array (1/(1+2*x^2), x/(1+2*x^2)).

Examples

			Triangle begins:
    1;
    0,   1;
    2,   0,   1;
    0,   4,   0,  1;
    8,   0,   6,  0,  1;
    0,  20,   0,  8,  0,  1;
   40,   0,  36,  0, 10,  0,  1;
    0, 112,   0, 56,  0, 12,  0, 1;
  224,   0, 224,  0, 80,  0, 14, 0, 1;
Production matrix begins as:
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  0, 0, 2, 0, 1;
  0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A064062, A126087 (row sums).

Programs

Mathematica

T[n_, k_]:= (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*Binomial[n+1, (n-k)/2];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

Sage

def A155761(n,k): return (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*binomial(n+1, (n-k)/2)
flatten([[A155761(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021

Formula

T(n,k) = (1+(-1)^(n-k)) * ((k+1)/(n+1)) * binomial(n+1, (n-k)/2) * 2^((n-k-2)/2).

Sum_{k=0..n} T(n, k) = A126087(n).

T(n,k) = 2^((n-k)/2) * A053121(n,k). - Philippe Deléham, Feb 11 2009

Sum_{k=0..n} T(2*n-k, k) = A064062(n+1). - G. C. Greubel, Jun 06 2021