A242566 Expansion of (1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x))/2.

0, 1, 1, 3, 7, 22, 67, 225, 765, 2704, 9710, 35558, 131859, 494892, 1874901, 7162807, 27558511, 106695148, 415346144, 1624780952, 6383671910, 25179642120, 99670897534, 395810459602, 1576464630375, 6295827843098

Offset: 0

Author

Vladimir Kruchinin, Jun 09 2014

Keywords

• nonn
• easy

Comments

The sequence 1, 1, 3, 7, ... with offset 0 is the Riordan transform with the Riordan matrix A053121 (the inverse of the Chebyshev S matrix A049310) of the Catalan sequence A000108. - Wolfdieter Lang, Feb 18 2017

Crossrefs

Cf. A000108, A049310, A053121, A101499.

Programs

Mathematica

CoefficientList[Series[1/2 - Sqrt[(-2 + x + 2*Sqrt[1-4*x^2])/x]/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)

Maxima

a(n):=sum(binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i),i,0,(n-1)/2)/(n);

PARI

a(n) = if (n, sum(i=0,(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, 0); \\ Michel Marcus, Jun 09 2014

Formula

a(n) = sum(i=0..(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, n>0, a(0)=0.

G.f. A(x) = x*C(x^2)*C(x*C(x^2)), where C(x) is g.f. A000108.

G.f. A(x) satisfies A(x)=x*(1/(1-A(x))+A(x)^2-A(x)^3).

a(n) ~ 17^(n+1/2) / (sqrt(15*Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Jun 15 2014

Conjecture D-finite with recurrence: 2*n*(2*n+1)*a(n) +(-49*n^2+97*n-36)*a(n-1) +12*(10*n^2-42*n+41)*a(n-2) +4*(49*n-97)*(n-3)*a(n-3) -544*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jan 25 2020