A375438 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.

1, 1, 1, 1, 2, 4, 6, 12, 24, 46, 93, 191, 393, 819, 1724, 3648, 7772, 16654, 35850, 77520, 168289, 366629, 801328, 1756620, 3861157, 8508247, 18791480, 41591566, 92237598, 204931918, 456096480, 1016720162, 2269865456, 5074732028, 11360680664, 25464831258, 57146836290

Offset: 1

Author

Paul D. Hanna, Aug 21 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 24*x^9 + 46*x^10 + 93*x^11 + 191*x^12 + 393*x^13 + 819*x^14 + 1724*x^15 + ...
where A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 36*x^8 + 70*x^9 + 138*x^10 + 279*x^11 + 571*x^12 + 1179*x^13 + 2457*x^14 + 5168*x^15 + ...
Let B(x) be the series reversion, B(A(x)) = x, then B(x) begins
B(x) = x - x^2 + x^3 - x^4 + 3*x^6 - 8*x^7 + 9*x^8 + 18*x^9 - 134*x^10 + 442*x^11 - 997*x^12 + 1428*x^13 - 10*x^14 - 7640*x^15 + ...
SPECIFIC VALUES.
A(2/5) = 0.741461459188681119672668058998130332678610537393868...
A(1/3) = 0.515838591521774544528452689654484632143493145820237...
A(1/4) = 0.335691575266570204286454430830296229544471248787335...
A(1/5) = 0.250623759777806277498640241328479184446996870792060...
A(1/6) = 0.200220725056085320106333620370114891484630832364434...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A375439, A357538, A000625.

Programs

PARI

{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
A[#A] = polcoeff( x + x^2 + ( Ax^3 + 2*subst(Ax,x,x^3))/3 - Ax,#A-1) );A[n+1]}
for(n=1,40,print1(a(n),", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.3394011548205191342723840863090815005163727107... and c = 0.278795592719328257510209001410993009380027455... - Vaclav Kotesovec, Aug 22 2024