A171991 G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^4.
1, 1, 4, 25, 190, 1644, 15636, 159977, 1736392, 19804646, 235758596, 2914539808, 37275052828, 491727695628, 6675266957376, 93073877729749, 1330851640325930, 19490214361291636, 292025342161401292, 4472427699080578554, 69958787931298226280, 1116911386287712897260
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1644*x^6 + 15636*x^7 +... Related series begin: A(A(x)) = x + 2*x^2 + 10*x^3 + 71*x^4 + 598*x^5 + 5634*x^6 + 57624*x^7 +... A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 182*x^5 + 1580*x^6 + 15080*x^7 +... A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 556*x^7 + 5192*x^8 + 51536*x^9 +... A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 446*x^6 + 3868*x^7 + 36705*x^8 +... A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 152*x^7 + 1205*x^8 + 10564*x^9 +... where the series reversion of the g.f. A(x) begins: x - A(x)^2 - A(x)^4 = x - x^2 - 2*x^3 - 10*x^4 - 62*x^5 - 468*x^6 - 4020*x^7 -...
Programs
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Mathematica
terms = 23; A[] = 0; Do[A[x] = x + A[A[x]]^2 + A[A[x]]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 04 2025 *)
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PARI
{a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2+B^4); polcoeff(A, n)} -
PARI
{a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^4+x*O(x^n))); polcoeff(A, n)} for(n=1,30,print1(a(n),", "))
Formula
G.f. satisfies: A( x - A(x)^2 - A(x)^4 ) = x.