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 A242566 Expansion of (1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x))/2. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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 LINKS Table of n, a(n) for n=0..25. 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 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 *) PROG (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 CROSSREFS Cf. A000108, A049310, A053121, A101499. Sequence in context: A148685 A365452 A035353 * A148686 A148687 A319312 Adjacent sequences: A242563 A242564 A242565 * A242567 A242568 A242569 KEYWORD nonn,easy AUTHOR Vladimir Kruchinin, Jun 09 2014 STATUS approved

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Last modified August 11 13:26 EDT 2024. Contains 375069 sequences. (Running on oeis4.)