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

0,4

Comments

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

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A064062, A126087 (row sums).

Sequence in context: A111959 A110109 A145973 * A067631 A134317 A123641

Adjacent sequences: A155758 A155759 A155760 * A155762 A155763 A155764

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 26 2009

Status

approved