A291744 G.f. A(x) satisfies: A(x - x*A(x)) = x + 2*x*A(x).

1, 3, 15, 105, 897, 8739, 93663, 1080909, 13246017, 170728251, 2298619851, 32162768805, 465875706873, 6964550221215, 107193366978651, 1695277029466917, 27504875620268325, 457183442035485927, 7776605660061178251, 135234473290510961097, 2402252449086179775861, 43557766261735276367055, 805650777590230815177879, 15191845940176304945626737, 291896599103455803872483709, 5712079123789080942126760083

Offset: 1

Author

Paul D. Hanna, Aug 30 2017

Keywords

• nonn

Examples

			G.f.: A(x) = x + 3*x^2 + 15*x^3 + 105*x^4 + 897*x^5 + 8739*x^6 + 93663*x^7 + 1080909*x^8 + 13246017*x^9 + 170728251*x^10 + 2298619851*x^11 + 32162768805*x^12 +...
such that A(x - x*A(x)) = x + 2*x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + 2*x^2 + 6*x^3 + 30*x^4 + 210*x^5 + 1794*x^6 + 17478*x^7 + 187326*x^8 + 2161818*x^9 + 26492034*x^10 + 341456502*x^11 + 4597239702*x^12 +...
which equals x + 2*x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 5*x^3 + 35*x^4 + 299*x^5 + 2913*x^6 + 31221*x^7 + 360303*x^8 + 4415339*x^9 + 56909417*x^10 + 766206617*x^11 + 10720922935*x^12 +...
which equals (2*x + A(x))/3.
A( (2*x + A(x))/3 ) = x + 4*x^2 + 26*x^3 + 218*x^4 + 2126*x^5 + 22986*x^6 + 268410*x^7 + 3331482*x^8 + 43492370*x^9 + 592851806*x^10 + 8393229602*x^11 + 122922601030*x^12 +...
which equals (A(x) - x) / (A(x) + 2*x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A291743, A276358.

Programs

PARI

{a(n) = my(A=x); for(i=1, n, A = 3*serreverse( x - x*A +x*O(x^n) ) - 2*x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=x, B); for(i=1, n, B = (2*x + A)/3 +x*O(x^n); A = x*(1 + 2*subst(A, x, B))/(1 - subst(A, x, B)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = 3 * Series_Reversion( x - x*A(x) ) - 2*x.

(2) A(x) = x * (1 + 2*A(B(x))) / (1 - A(B(x))), where B(x) = (2*x + A(x))/3.

(3) A( (2*x + A(x))/3 ) = (A(x) - x) / (A(x) + 2*x).