OFFSET
0,4
COMMENTS
Row sums are A098615.
Binomial transform (product with C(n,k)) is A111960.
Diagonal sums are A026671 (with interpolated zeros).
Inverse is (1/sqrt(1+4x^2),x/sqrt(1+4x^2)), or (sqrt(-1))^(n-k)*T(n,k). [corrected by Peter Bala, Aug 13 2021]
The Riordan array (1,x/sqrt(1-4*x^2)) is the same array with an additional column of zeros (besides the top element 1) added to the left. - Vladimir Kruchinin, Feb 17 2011
LINKS
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
FORMULA
Riordan array (1/sqrt(1-4x^2), x/sqrt(1-4x^2)); number triangle T(n, k)=(1+(-1)^(n-k))*binomial((n-1)/2, (n-k)/2)*2^(n-k)/2.
G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Peter Bala, Aug 13 2021: (Start)
The row entries, read from right to left, are the coefficients in the n-th order Taylor polynomial of (sqrt(1 + 4*x^2))^((n-1)/2) at x = 0.
The infinitesimal generator of this array has the sequence [2, 4, 6, 8, 10, ...] on the second subdiagonal below the main diagonal and zeros elsewhere.
The m-th power of the array is the Riordan array (1/sqrt(1 - 4*m*x^2), x/sqrt(1 - 4*m*x^2)) with entries given by sqrt(m)^(n-k)*T(n,k). (End)
EXAMPLE
From Peter Bala, Aug 13 2021: (Start)
Triangle begins
1;
0, 1;
2, 0, 1;
0, 4, 0, 1;
6, 0, 6, 0, 1;
0, 16, 0, 8, 0, 1;
Infinitesimal generator begins
0;
0, 0;
2, 0, 0;
0, 4, 0, 0;
0, 0, 6, 0, 0;
0, 0, 0, 8, 0, 0; (End)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Aug 23 2005
STATUS
approved