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A192730
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G.f. satisfies: A(x) = 1/(1 - x*A(x)^3/(1 - x^2*A(x)^3/(1 - x^3*A(x)^3/(1 - x^4*A(x)^3/(1 - ...))))), a recursive continued fraction.
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5
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1, 1, 4, 23, 151, 1075, 8075, 62996, 505501, 4145684, 34594540, 292794156, 2507383158, 21686318745, 189162110341, 1662142617881, 14698913545378, 130723572694407, 1168419986539867, 10490326933563842, 94564400499455397, 855552893388047193
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: A(x) = P(x)/Q(x) where
_ P(x) = Sum_{n>=0} x^(n*(n+1)) * (-A(x)^3)^n / Product(k=1..n} (1-x^k),
_ Q(x) = Sum_{n>=0} x^(n^2) * (-A(x)^3)^n / Product(k=1..n} (1-x^k),
due to Ramanujan's continued fraction identity.
a(n) ~ c * d^n / n^(3/2), where d = 9.72359087408044730447308019524191930733163... and c = 0.151620024312256318854728680725808488795... - Vaclav Kotesovec, Nov 18 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 151*x^4 + 1075*x^5 + 8075*x^6 +...
which satisfies A(x) = P(x)/Q(x) where
P(x) = 1 - x^2*A(x)^3/(1-x) + x^6*A(x)^6/((1-x)*(1-x^2)) - x^12*A(x)^9/((1-x)*(1-x^2)*(1-x^3)) + x^20*A(x)^12/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Q(x) = 1 - x*A(x)^3/(1-x) + x^4*A(x)^6/((1-x)*(1-x^2)) - x^9*A(x)^9/((1-x)*(1-x^2)*(1-x^3)) + x^16*A(x)^12/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Explicitly, the above series begin:
P(x) = 1 - x^2 - 4*x^3 - 19*x^4 - 113*x^5 - 763*x^6 - 5557*x^7 - 42472*x^8 - 335804*x^9 - 2723164*x^10 - 22523476*x^11 - 189267247*x^12 +...
Q(x) = 1 - x - 4*x^2 - 19*x^3 - 112*x^4 - 757*x^5 - 5517*x^6 - 42188*x^7 - 333673*x^8 - 2706555*x^9 - 22390279*x^10 - 188175369*x^11 - 1602132261*x^12 +...
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PROG
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(PARI) /* As a recursive continued fraction: */
{a(n)=local(A=1+x, CF); for(i=1, n, CF=1+x; for(k=0, n, CF=1/(1-x^(n-k+1)*A^3*CF+x*O(x^n))); A=CF); polcoeff(A, n)}
(PARI) /* By Ramanujan's continued fraction identity: */
{a(n)=local(A=1+x, P, Q); for(i=1, n,
P=sum(m=0, sqrtint(n), x^(m*(m+1))/prod(k=1, m, 1-x^k)*(-A^3+x*O(x^n))^m);
Q=sum(m=0, sqrtint(n), x^(m^2)/prod(k=1, m, 1-x^k)*(-A^3+x*O(x^n))^m); A=P/Q); polcoeff(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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