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
KEYWORD
AUTHOR
Paul Barry, Jan 26 2009
STATUS
approved