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A274734
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G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x*A(x)^2).
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4
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1, 3, 15, 94, 661, 4983, 39363, 321587, 2694860, 23035341, 200068651, 1760558682, 15663027711, 140648129383, 1273083938979, 11603500739475, 106404140837773, 980977232554344, 9087285865886766, 84541177049414342, 789545725457924023, 7399515198155161271, 69568021610270590583, 655960254857760518109, 6201585037793334756198, 58775103307105512895151
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OFFSET
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0,2
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COMMENTS
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More generally, if G(x) satisfies
G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).
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LINKS
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FORMULA
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G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^2) / (1+x)^2 ).
Recurrence: 31*(n-1)*n*(n+1)*(5974*n^3 - 40359*n^2 + 90115*n - 67124)*a(n) = 2*(n-1)*n*(1003632*n^4 - 7282128*n^3 + 18518502*n^2 - 18822839*n + 5649607)*a(n-1) - 2*(n-1)*(740776*n^5 - 6486068*n^4 + 21715762*n^3 - 34616651*n^2 + 26123385*n - 7413210)*a(n-2) + 2*(2*n - 5)*(65714*n^5 - 575377*n^4 + 1957337*n^3 - 3264653*n^2 + 2726129*n - 941430)*a(n-3) + 4*(n-3)*(2*n - 7)*(2*n - 5)*(5974*n^3 - 22437*n^2 + 27319*n - 11394)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((s*(1+r*s)*(2 + s + 3*r*s^2)) / (1 + r*(1 + 6*s*(1+r*s)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.099424837262345547872398211374352678... and s = 2.183663565361369673488934371066403742... are roots of the system of equations (1 + r*s)^2*(1 + r*s^2) = s, 2*r*(1 + s + 2*r^2*s^3 + r*s*(1 + 3*s)) = 1. - Vaclav Kotesovec, Nov 18 2017
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+2*k+2,n-k). - Seiichi Manyama, Jan 27 2024
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 15*x^2 + 94*x^3 + 661*x^4 + 4983*x^5 + 39363*x^6 + 321587*x^7 +...
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^2 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A, n)}
for(n=0, 30, 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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