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%I #38 Jul 24 2023 09:22:20
%S 1,1,4,18,88,456,2464,13736,78432,456416,2697088,16141120,97632000,
%T 595912960,3665728512,22703097472,141448381952,885934151168,
%U 5575020435456,35230798994432,223485795258368,1422572226146304,9083682419818496,58169612565614592,373486362257899520,2403850703479816192
%N Generalized Catalan numbers.
%C a(n) = K(2,2; n)/2 with K(a,b; n) defined in a comment to A068763.
%C Hankel transform is A166232(n+1). - _Paul Barry_, Oct 09 2009
%H Vincenzo Librandi, <a href="/A068764/b068764.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-sqrt(1-8*x*(1-x)))/(4*x).
%F a(n+1) = 2*sum(a(k)*a(n-k), k=0..n), n>=1, a(0) = 1 = a(1).
%F a(n) = (2^n)*p(n, -1/2) with the row polynomials p(n, x) defined from array A068763.
%F E.g.f. (offset -1) is exp(4*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - _Vladeta Jovovic_, Mar 31 2004
%F The o.g.f. satisfies A(x) = 1 + x*(2*A(x)^2 - 1), A(0) = 1. - _Wolfdieter Lang_, Nov 13 2007
%F a(n) = subs(t=1,(d^(n-1)/dt^(n-1))(-1+2*t^2)^n)/n!, n >= 2, due to the Lagrange series for the given implicit o.g.f. equation. This formula holds also for n=1 if no differentiation is used. - _Wolfdieter Lang_, Nov 13 2007, Feb 22 2008
%F 1/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-..... (continued fraction). - _Paul Barry_, Jan 29 2009
%F a(n) = A166229(n)/(2-0^n). - _Paul Barry_, Oct 09 2009
%F a(n) = sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n), n>0. - _Vladimir Kruchinin_, Aug 11 2010
%F D-finite with recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012
%F a(n) = 4^(n-1)*hypergeom([(1-n)/2,1-n/2], [2], 1/2) + 0^n/sqrt(2). - _Vladimir Reshetnikov_, Nov 07 2015
%F 0 = a(n)*(+64*a(n+1) - 160*a(n+2) + 32*a(n+3)) + a(n+1)*(+32*a(n+1) + 48*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n>=0. - _Michael Somos_, Nov 08 2015
%F a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(2*k+1,n) / (2*k+1). - _Seiichi Manyama_, Jul 24 2023
%e G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ...
%t Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)
%t Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 07 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* _Michael Somos_, Nov 08 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* _Michael Somos_, Mar 31 2007 */
%o (Maxima) a(n):=sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n); \\ _Vladimir Kruchinin_, Aug 11 2010
%o (PARI) x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ _Joerg Arndt_, May 06 2013
%Y Cf. A025227, A025228, A025229, A025230.
%Y Cf. A071356, A001003, A025235.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002
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