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A186996
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G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^4.
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7
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1, 1, 2, 6, 20, 72, 273, 1073, 4333, 17869, 74937, 318601, 1370113, 5949201, 26046727, 114857599, 509669295, 2274146599, 10197234215, 45925646367, 207656685443, 942302814363, 4289903653615, 19588180438263, 89685571667763
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OFFSET
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0,3
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COMMENTS
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Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.
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LINKS
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FORMULA
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G.f.: A(x) = (1/x)*Series_Reversion( 2*x^3/(1 - sqrt(1-4*x^2-4*x^3)) ).
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^3) / (1 + x*A(x)^2).
Recurrence: 3*(n-2)*n*(2*n-5)*(3*n-2)*(3*n+2)*a(n) = 3*(2*n-5)*(2*n-1)*(18*n^3 - 54*n^2 + 36*n - 5)*a(n-1) + (2*n-3)*(94*n^4 - 564*n^3 + 1222*n^2 - 1128*n + 331)*a(n-2) + 3*(2*n - 5)*(2*n - 1)*(18*n^3 - 108*n^2 + 198*n - 103)*a(n-3) - 3*(n-3)*(n-1)*(2*n-1)*(3*n-11)*(3*n-7)*a(n-4). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = (9 + 8*sqrt(3) + 4*sqrt(12 + 9*sqrt(3)))/9 = 4.874032512954972962... is the root of the equation 27 - 108*d - 94*d^2 - 108*d^3 + 27*d^4 = 0 and c = sqrt(11/(8*(-44 + sqrt(748 + 891*sqrt(3)))*Pi)) = 0.336422381089368230542882135982348331566666028... - Vaclav Kotesovec, Sep 10 2013, updated Apr 27 2015
a(n) = sum(k=0..n, binomial(3n-2k+1, k)*binomial(k,n-k)*1/(3n-2k+1)). - Michael D. Weiner, Apr 23 2015
G.f.: A(x) = hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x^2/(x-1)^4)/(1-x). - Robert Israel, Apr 28 2015
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 273*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 56*x^4 + 208*x^5 + 806*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 114*x^4 + 438*x^5 + 1739*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 201*x^4 + 800*x^5 + 3260*x^6 +...
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MAPLE
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S:= series(RootOf(y - 1 - x*y - x^2*y^4, y), x, 101):
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MATHEMATICA
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nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF + x^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) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A+x^2*(A+x*O(x^n))^4); polcoeff(A, n)}
(PARI) {a(n)=polcoeff((1/x)*serreverse(2*x^3/(1 - sqrt(1-4*x^2-4*x^3 +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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