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A199877
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G.f. satisfies: A(x) = (1 + x*A(x)^4)*(1 + x^2*A(x)^4).
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15
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1, 1, 5, 31, 222, 1727, 14179, 120930, 1060992, 9514463, 86818391, 803516167, 7524700644, 71169939341, 678877680077, 6523424076116, 63087757216084, 613575943566436, 5997490784042496, 58886692596764215, 580516324380845804, 5743718741275361697, 57017511243375535969
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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. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k] * x^n*A(x)^(3*n)/n ).
(2) A(x) = exp( Sum_{n>=1} [(1-x)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k )] * x^n*A(x)^(3*n)/n.
a(n) ~ s * sqrt((1 + 2*r + 3*r^2*s^4) / (2*Pi*(3 + 3*r + 14*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.0940387024218615638441791629908854357421782432118... and s = 1.322930427586092521664829345633697493713415726621... are real roots of the system of equations 1 + r*(1 + r)*s^4 + r^3*s^8 = s, 4*r*s^3*(1 + r + 2*r^2*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 222*x^4 + 1727*x^5 + 14179*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 26*x^2 + 188*x^3 + 1471*x^4 + 12124*x^5 + 103684*x^6 +...
A(x)^8 = 1 + 8*x + 68*x^2 + 584*x^3 + 5122*x^4 + 45792*x^5 + 416196*x^6 +...
where A(x) = 1 + x*(1+x)*A(x)^4 + x^3*A(x)^8.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x*A(x)^3 + (1 + 2^2*x + x^2)*x^2*A(x)^6/2 +
(1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^9/3 +
(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^12/4 +
(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^15/5 +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^6*A(x)^18/6 +...
more explicitly,
log(A(x)) = x + 9*x^2/2 + 79*x^3/3 + 733*x^4/4 + 7006*x^5/5 + 68229*x^6/6 + 673268*x^7/7 +...
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MATHEMATICA
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terms = 23; A[_] = 1; Do[A[x_] = (1 + x*A[x]^4)*(1 + x^2*A[x]^4) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 09 2018 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A^4)*(1 + x^2*A^4)+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j)*(x*A^3+x*O(x^n))^m/m))); polcoeff(A, n, x)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j)*x^m*A^(3*m)/m))); polcoeff(A, n, x)}
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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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