A276361 G.f. satisfies: A(x - 2*A(x)^2) = x + A(x)^2.

1, 3, 30, 447, 8202, 171846, 3956796, 97916895, 2567551890, 70655670690, 2026596875268, 60282027684678, 1852444347792036, 58633762133405100, 1907098496516434680, 63620675921801106495, 2173457638433471757282, 75940916632597398212298, 2710857429948875567968692, 98775527832178103444182722, 3670845430153146908693608044, 139047871842184594320103381524, 5365224711989826990651317756232

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) also satisfies:

(1) A(x) = x + 3 * A( x/3 + 2*A(x)/3 )^2.

(2) 2*A(x) = -x + 3 * Series_Reversion(x - 2*A(x)^2).

(3) R(x) = -2*x + 3 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.

(4) R( sqrt( x/3 - R(x)/3 ) ) = 2*x/3 + R(x)/3, where R(A(x)) = x.

a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k*3^(n-k-1).

Example

G.f.: A(x) = x + 3*x^2 + 30*x^3 + 447*x^4 + 8202*x^5 + 171846*x^6 + 3956796*x^7 + 97916895*x^8 + 2567551890*x^9 + 70655670690*x^10 + 2026596875268*x^11 + 60282027684678*x^12 +...
such that A(x - 2*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = x/3 + 2*A(x)/3, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5468*x^5 + 114564*x^6 + 2637864*x^7 + 65277930*x^8 + 1711701260*x^9 + 47103780460*x^10 + 1351064583512*x^11 + 40188018456452*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 12*x^3 - 132*x^4 - 1992*x^5 - 36144*x^6 - 742176*x^7 - 16688880*x^8 - 402824928*x^9 - 10300868160*x^10 - 276531035520*x^11 - 7742210941056*x^12 +...
then Series_Reversion(x + A(x)^2) = 2*x/3 + R(x)/3.

Mathematica

m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[x/3 + 2 A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)

Prog

(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-2*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276362, A276363.

Sequence in context: A007004 A354291 A354287 * A360973 A359972 A218304

Adjacent sequences: A276358 A276359 A276360 * A276362 A276363 A276364

Keyword

nonn

Author

Paul D. Hanna, Aug 31 2016

Status

approved