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A192945
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G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n.
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6
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1, 1, 2, 9, 50, 311, 2072, 14460, 104346, 772255, 5829538, 44710705, 347424376, 2729299748, 21640457360, 172957598120, 1391926695402, 11270059892943, 91740990170150, 750364940281275, 6163650579487170, 50824871829196575
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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. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
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LINKS
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FORMULA
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G.f. A(x) equals the formal inverse of function (x-1)*(2-x^2).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^2).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^2 ).
a(n) = (1/n)*Sum_{k=0..n-1} binomial(n+k-1, n-1)*Sum_{i=ceiling((n-k-1)/2)..n-k-1} binomial(i, n-k-i-1)*binomial(n+k+i-1, n+k-1), n > 0, a(0)=1. - Vladimir Kruchinin, Oct 11 2011
Recurrence: 8*(n-1)*n*a(n) = 34*(n-1)*(2*n-3)*a(n-1) + 3*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Nov 20 2012
a(n) ~ 1/3*sqrt(7/6 - 17/(6*sqrt(7)))*((17 + 7*sqrt(7))/4)^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 20 2012
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 50*x^4 + 311*x^5 + 2072*x^6 + ...
where (A(x) - 1)*(2 - A(x)^2) = x
and A(x - 2*x^2 - x^3) = 1 + x.
Related expansions:
(A(x)^2-1) = 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
(A(x)^2-1)^2 = 4*x^2 + 20*x^3 + 113*x^4 + 708*x^5 + 4736*x^6 + ...
(A(x)^2-1)^3 = 8*x^3 + 60*x^4 + 414*x^5 + 2909*x^6 + 20970*x^7 + ...
(A(x)^2-1)^4 = 16*x^4 + 160*x^5 + 1304*x^6 + 10184*x^7 + ...
Also,
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 222*x^4 + 1380*x^5 + 9191*x^6 + ...
where 2 + x = 2*A(x) + A(x)^2 - A(x)^3.
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MATHEMATICA
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Flatten[{1, Table[1/n*Sum[Binomial[n+k-1, n-1]*Sum[Binomial[i, n-k-i-1]*Binomial[n+k+i-1, n+k-1], {i, Floor[(n-k-1)/2], n-k-1}], {k, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec after Vladimir Kruchinin, Nov 20 2012 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^2-1+x*O(x^n))^m)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^2+x^2*O(x^n))); polcoeff(A, n)}
(Maxima) a(n):=if n=0 then 1 else 1/n*sum(binomial(n+k-1, n-1) *sum(binomial(i, n-k-i-1)*binomial(n+k+i-1, n+k-1), i, ceiling((n-k-1)/2), n-k-1), k, 0, n-1); /* Vladimir Kruchinin, Oct 11 2011 */
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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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