A302701 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.
1, 1, 3, 16, 118, 1050, 10509, 113892, 1307043, 15661024, 194075098, 2470848492, 32161635070, 426440290744, 5743575712131, 78405535427220, 1082876597440146, 15109514661352482, 212736976140479073, 3019422091269739704, 43164665664066028062, 621078277521084894978, 8989001884449529431990, 130795752983608734209604, 1912460927749734257739153, 28088780052768915388505436, 414247711043291214286003410
Offset: 0
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1050*x^5 + 10509*x^6 + 113892*x^7 + 1307043*x^8 + 15661024*x^9 + 194075098*x^10 + ... RELATED SERIES. (x/A(x))' / (x/A(x)^4)' = 1 + 6*x + 48*x^2 + 472*x^3 + 5250*x^4 + 63054*x^5 + 797244*x^6 + 10456344*x^7 + 140949216*x^8 + 1940750980*x^9 + ... which equals A'(x). The logarithmic derivative of the g.f. begins: A'(x)/A(x) = 1 + 5*x + 40*x^2 + 401*x^3 + 4531*x^4 + 55040*x^5 + 701716*x^6 + 9261257*x^7 + 125449600*x^8 + 1734071855*x^9 + 24362189248*x^10 + ... which equals (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4))/(8*x).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..400
Programs
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Mathematica
nmax = 30; A = 1; Do[A = 1 + Integrate[D[x/A, x]/D[x/A^4, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] (* Vaclav Kotesovec, Oct 15 2020 *)
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PARI
{a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A)'/(x/A^4 +x*O(x^n))' ); ); polcoeff(A, n)} for(n=0,30,print1(a(n),", "))
Formula
O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) - x*A'(x)) / (A(x) - 4*x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4) )/(8*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 + x*A(x)^2)*A'(x) + 4*x*A'(x)^2.
a(n) ~ 3^(2/3) * (1240209 - 716035*sqrt(3))^(1/6) * 2^((4*n - 5)/3) * (3 + 2*sqrt(3))^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Oct 14 2020