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A244062
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G.f. satisfies: A(x) = Sum_{n>=0} x^n * (2*A(x)^(2*n) - A(x)^n).
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1
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1, 1, 4, 21, 124, 786, 5228, 36005, 254568, 1837214, 13479308, 100239418, 753880440, 5724153044, 43820345784, 337850230061, 2621033435856, 20445810352950, 160271222750348, 1261838520251886, 9973780991950168, 79115475268744044, 629605388017281768, 5025263773704414050
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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) = 2/(1 - x*A(x)^2) - 1/(1 - x*A(x)).
G.f. satisfies: A(x) = G( x*(1-x*A(x)) / (1-2*x*A(x)) ) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
G.f.: A(x) = x/Series_Reversion(x*F(x)) where A(x) = F(x*A(x)) and F(x) = (1-2*x - sqrt((1-2*x)*(1-6*x+4*x^2)))/(2*x*(1-x)) is the g.f. of A059278.
Recurrence: n*(n+1)*(2*n+1)*(2015*n^7 - 27625*n^6 + 141203*n^5 - 313033*n^4 + 208928*n^3 + 209582*n^2 - 249690*n - 1260)*a(n) = 2*n*(10075*n^9 - 138125*n^8 + 689505*n^7 - 1342581*n^6 - 90999*n^5 + 3620601*n^4 - 3189085*n^3 - 1422829*n^2 + 2448474*n - 629856)*a(n-1) + 4*(28210*n^10 - 429065*n^9 + 2551767*n^8 - 7248183*n^7 + 8762631*n^6 + 1255293*n^5 - 12916972*n^4 + 8733443*n^3 + 1573824*n^2 - 2841228*n + 619920)*a(n-2) + 12*(12090*n^10 - 202020*n^9 + 1404463*n^8 - 5288640*n^7 + 11807054*n^6 - 15771984*n^5 + 9953477*n^4 + 5954400*n^3 - 18326984*n^2 + 13162824*n - 2348640)*a(n-3) - 8*(n-3)*(88660*n^9 - 1348490*n^8 + 7652292*n^7 - 18093912*n^6 + 5534031*n^5 + 47300904*n^4 - 60179995*n^3 - 14703742*n^2 + 49764312*n - 13965840)*a(n-4) + 24*(n-4)*(3*n - 14)*(3*n - 13)*(2015*n^7 - 13520*n^6 + 17768*n^5 + 49132*n^4 - 113149*n^3 - 1862*n^2 + 98496*n - 29880)*a(n-5). - Vaclav Kotesovec, Jun 19 2014
a(n) ~ sqrt(s*(1/r - 1/(1-r*s)^2) / (2*(1+3*r*s^2)/(1-r*s^2)^3 - r/(1-r*s)^3)) / (2*sqrt(2*Pi) * n^(3/2) * r^n), where r = 0.1173880603216979159912271683495821643016169... and s = 1.4338257350727901441223535965394946191082412... are roots of the system of equations 1/(r*s-1) + 2/(1-r*s^2) = s, 4*r*s/(r*s^2-1)^2 = 1+r/(r*s-1)^2. - Vaclav Kotesovec, Jun 19 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 124*x^4 + 786*x^5 + 5228*x^6 +...
where
A(x) = 1 + x*(2*A(x)^2 - A(x)) + x^2*(2*A(x)^4 - A(x)^2) + x^3*(2*A(x)^6 - A(x)^3) + x^4*(2*A(x)^8 - A(x)^4) + x^5*(2*A(x)^10 - A(x)^5) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(2*A^(2*m) - A^m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=2/(1-x*A^2 +x*O(x^n)) - 1/(1-x*A +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x, A=1+x); for(i=1, n, G=1+x*G^3+x*O(x^n)); for(i=1, n, A=subst(G, x, x*(1-x*A)/(1-2*x*A)) ); 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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