%I #29 Aug 31 2021 02:56:38

%S 1,0,1,2,0,1,0,4,0,1,6,0,6,0,1,0,16,0,8,0,1,20,0,30,0,10,0,1,0,64,0,

%T 48,0,12,0,1,70,0,140,0,70,0,14,0,1,0,256,0,256,0,96,0,16,0,1,252,0,

%U 630,0,420,0,126,0,18,0,1,0,1024,0,1280,0,640,0,160,0,20,0,1,924,0,2772,0

%N Renewal array for aerated central binomial coefficients.

%C Row sums are A098615.

%C Binomial transform (product with C(n,k)) is A111960.

%C Diagonal sums are A026671 (with interpolated zeros).

%C 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]

%C 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

%H Paul Barry, <a href="https://arxiv.org/abs/2101.10218">On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths</a>, arXiv:2101.10218 [math.CO], 2021.

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F 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.

%F G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - _Paul Barry_, Jan 28 2009

%F From _Peter Bala_, Aug 13 2021: (Start)

%F T(2*n,2*k) = A046521(n,k); T(2*n+1,2*k+1) = A038231(n,k).

%F 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.

%F 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.

%F 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)

%e From _Peter Bala_, Aug 13 2021: (Start)

%e Triangle begins

%e 1;

%e 0, 1;

%e 2, 0, 1;

%e 0, 4, 0, 1;

%e 6, 0, 6, 0, 1;

%e 0, 16, 0, 8, 0, 1;

%e Infinitesimal generator begins

%e 0;

%e 0, 0;

%e 2, 0, 0;

%e 0, 4, 0, 0;

%e 0, 0, 6, 0, 0;

%e 0, 0, 0, 8, 0, 0; (End)

%Y Cf. A054335, A038231, A046521.

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Aug 23 2005