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A199475
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G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).
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23
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1, 2, 7, 34, 195, 1225, 8146, 56336, 401005, 2918308, 21614216, 162385693, 1234515922, 9479336440, 73410868630, 572719097908, 4496923141241, 35509806367132, 281814387290431, 2246608404455588, 17982234787607464, 144458551104066553, 1164342291135424494
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OFFSET
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0,2
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COMMENTS
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Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):
B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1/((1-x)*(1 - x*A(x)^2)).
G.f.: A(x) = (1/x)*Series_Reversion( 2*x^2/(1+x^2 - sqrt(1-4*x-2*x^2+x^4)) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) = g.f. of A171199, where G(x) = exp( Sum_{n>=1} [G(x)^n + G(x)^-n]*x^n/n ).
a(n) = 1 + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 25 2021
a(n) ~ sqrt(387 + 35*sqrt(129)) * (35 + 3*sqrt(129))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Jul 30 2021
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023
D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(-13*n^2+11*n-2)*a(n-1) +(35*n^2-23*n-42)*a(n-2) +(35*n^2-257*n+426)*a(n-3) +3*(-13*n^2+93*n-166)*a(n-4) +2*(n-4)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Feb 10 2024
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 7*x^2 + 34*x^3 + 195*x^4 + 1225*x^5 +...
where g.f. A = A(x) satisfies the equivalent expressions:
A = 1 + x*(1-A^4)/(1-A^2) + x^2*(1-A^6)/(1-A^2) + x^3*(1-A^8)/(1-A^2) +...
A = 1 + x*(1 + A^2) + x^2*(1 + A^2 + A^4) + x^3*(1 + A^2 + A^4 + A^6) +...
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[(2*x^2)/(1 + x^2 - Sqrt[1 - 4*x - 2*x^2 + x^4]), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Jul 30 2021 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, A^(2*k))+x*O(x^n))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/((1-x)*(1 - x*A^2+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=polcoeff(1/x*serreverse(2*x^2/(1+x^2-sqrt(1-4*x-2*x^2+x^4+x^3*O(x^n)))), n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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