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A219534
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G.f. satisfies A(x) = 1 + x*(A(x)^2 + A(x)^4).
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12
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1, 2, 12, 100, 968, 10208, 113792, 1318832, 15732064, 191878592, 2381917824, 29995598208, 382257383168, 4920505410816, 63882881030656, 835554927932160, 10999486798112256, 145626782310460416, 1937772463214168064, 25901381584638605312, 347618773649248088064
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OFFSET
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0,2
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LINKS
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FORMULA
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Let G(x) = (1 - sqrt(1-4*x-4*x^2))/(2*x), then g.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion(x/G(x)^2) ),
(2) A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2),
where x*G(x) is the g.f. of A025227.
Recurrence: 3*n*(3*n-1)*(3*n+1)*(131*n^3 - 666*n^2 + 1075*n - 558)*a(n) = 2*(26200*n^6 - 172500*n^5 + 431572*n^4 - 521613*n^3 + 316327*n^2 - 89058*n + 8640)*a(n-1) - 12*(n-2)*(1441*n^5 - 8767*n^4 + 19186*n^3 - 18172*n^2 + 6930*n - 810)*a(n-2) + 8*(n-3)*(n-2)*(2*n-5)*(131*n^3 - 273*n^2 + 136*n - 18)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 2/81*(7217783 + 10611 * sqrt(786))^(1/3) + 74654/(81*(7217783 + 10611 * sqrt(786))^(1/3)) + 400/81 = 14.48001092254652246... is the root of the equation -16 + 132*d - 400*d^2 + 27*d^3 = 0 and c = 1/2358*sqrt(262)*sqrt((213070976 + 3034746 * sqrt(786))^(1/3) * ((213070976 + 3034746 * sqrt(786))^(2/3) + 336670 + 1310*(213070976 + 3034746 * sqrt(786))^(1/3)))/((213070976 + 3034746 * sqrt(786))^(1/3)*sqrt(Pi)) = 0.1929450901182412149... - Vaclav Kotesovec, Sep 10 2013
a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+1,n)/(2*n+2*k+1). - Seiichi Manyama, Apr 03 2024
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 968*x^4 + 10208*x^5 +...
Related expansions:
A(x)^2 = 1 + 4*x + 28*x^2 + 248*x^3 + 2480*x^4 + 26688*x^5 +...
A(x)^4 = 1 + 8*x + 72*x^2 + 720*x^3 + 7728*x^4 + 87104*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) where
G(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 144*x^5 + 544*x^6 +...+ A025227(n+1)*x^n +...
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MATHEMATICA
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nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[1+x*(AGF^2+AGF^4)-AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
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PROG
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(PARI) /* Formula A(x) = 1 + x*(A(x)^2 + A(x)^4): */
{a(n)=local(A=1); for(i=1, n, A=1+x*(A^2+A^4) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula using Series Reversion: */
{a(n)=local(A=1, G=(1-sqrt(1-4*x-4*x^2+x^3*O(x^n)))/(2*x)); A=sqrt((1/x)*serreverse(x/G^2)); polcoeff(A, n)}
for(n=0, 25, print1(a(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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