A290958 G.f. A(x) satisfies: A( 2*A(x)^2 + 4*A(x)^3 ) = 2*x^2 - 4*x^3.

1, -2, 6, -26, 100, -460, 2258, -11558, 60786, -326826, 1785930, -9893778, 55447800, -313817720, 1791442406, -10303155322, 59642852324, -347233450156, 2031756438046, -11941773701426, 70471288256196, -417379686511812, 2480161711278070, -14781955283569090, 88343937381017274, -529319474378769346, 3178848917169132254, -19131855254581689246

Offset

1,2

Comments

Series reversion of the g.f. is described by A290957.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 6.36304571910819028529344... and c = 0.086619593102483539978... - Vaclav Kotesovec, Aug 28 2017

Example

G.f.: A(x) = x - 2*x^2 + 6*x^3 - 26*x^4 + 100*x^5 - 460*x^6 + 2258*x^7 - 11558*x^8 + 60786*x^9 - 326826*x^10 + 1785930*x^11 - 9893778*x^12 + 55447800*x^13 - 313817720*x^14 + 1791442406*x^15 - 10303155322*x^16 + 59642852324*x^17 - 347233450156*x^18 + 2031756438046*x^19 - 11941773701426*x^20 +...
such that A( 2*A(x)^2 - 4*A(x)^3 ) = 2*x^2 + 4*x^3.
Let B(x) be the series reversion of A(x), then B(x) is the g.f. of A290957 and begins;
B(x) = x + 2*x^2 + 2*x^3 + 6*x^4 + 40*x^5 + 208*x^6 + 798*x^7 + 3122*x^8 + 15038*x^9 + 77830*x^10 + 381798*x^11 + 1819998*x^12 + 8925172*x^13 + 45280900*x^14 + 231030138*x^15 + 1171823534*x^16 +...+ A290957(n)*x^n +...
where B( 2*x^2 - 4*x^3 ) = 2*A(x)^2 + 4*A(x)^3,
also, A( 2*x^2 + 4*x^3 ) = 2*B(x)^2 - 4*B(x)^3,
and B( 2*B(x)^2 - 4*B(x)^3 ) = 2*x^2 + 4*x^3.
Related series begin:
2*A(x)^2 + 4*A(x)^3 = 2*x^2 + 12*x^3 + 40*x^4 + 112*x^5 + 416*x^6 + 2112*x^7 + 10336*x^8 + 45936*x^9 + 206192*x^10 + 999376*x^11 + 5026640*x^12 +...
2*B(x)^2 - 4*B(x)^3 = 2*x^2 - 12*x^3 + 56*x^4 - 272*x^5 + 1312*x^6 - 6432*x^7 + 32640*x^8 - 170576*x^9 + 911696*x^10 - 4963760*x^11 + 27425200*x^12 +...

Prog

(PARI) /* Informal code to generate N terms */
{C=[1, -2]; for(i=1, N=60,
A = sum(n=1, #C, C[n]*x^(n) ) + t*x^(#C+1)   +O(x^(#C+2));
S = subst(A, x, 2*A^2 + 4*A^3);
C = concat(C, polcoeff(subst(-S/deriv(polcoeff(S, #C+2, x), t), t, 0), #C+2, x) )); C}

Crossrefs

Cf. A290957 (inverse), A271961.

Sequence in context: A083845 A027239 A191821 * A323265 A285024 A192403

Adjacent sequences: A290955 A290956 A290957 * A290959 A290960 A290961

Keyword

sign

Author

Paul D. Hanna, Aug 14 2017

Status

approved