A383563 G.f. A(x) satisfies A( x*(1+x)/A(x)^2 ) = 1 + x.

1, 1, 1, 3, 13, 72, 465, 3362, 26531, 224856, 2024188, 19202830, 190857879, 1978567663, 21319434418, 238109360460, 2750229390071, 32789591062124, 402891169846242, 5094855923807780, 66229610059651788, 884081025776797026, 12107164229698851942, 169954380180177899277, 2443554376412586234247

Offset: 0

Author

Paul D. Hanna, May 26 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 72*x^5 + 465*x^6 + 3362*x^7 + 26531*x^8 + 224856*x^9 + 2024188*x^10 + ...
 where A( x*(1+x)/A(x)^2 ) = 1 + x.
RELATED SERIES.
The g.f. of A145345 begins
B(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + 10706*x^7 + 89120*x^8 + 794347*x^9 + ...
 where B(x/B(x)) = 1 + x*B(x)
 also, B( x/A(x)^2 ) = 1 + x.
The g.f. of A121687 begins
C(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + 37244*x^7 + 335153*x^8 + 3194510*x^9 + ...
 where C(x) = 1/(1 - x*C(x*C(x))^2)
 also, C( (x/(1+x))/A(x)^2 ) = 1 + x.
C(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 + 89120*x^7 + 794347*x^8 + ...
 where B(x) = 1 + x*C(x)^2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..470

Crossrefs

Cf. A384265, A145345 (B(x)), A121687 (C(x)).

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + x*serreverse(x/Ser(A)^2),#A) ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145345 and C(x) = g.f. of A121687 satisfies the following formulas.

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

(2.a) Series_Reversion( x/A(x) ) = x + x*Series_Reversion( x/A(x)^2 ).

(2.b) [x^n] A(x)^(n+1)/(n+1) = [x^(n-1)] A(x)^(2*n)/n for n >= 1.

(2.c) B(x) = 1 + x*C(x)^2.

(3.a) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C(x/B(x)).

(3.b) A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) = B(x*C(x)).

(4.a) A(x) = A(x)^2 - x*C(x/A(x))^2.

(4.b) B( x/A(x)^2 ) = 1 + x.

(4.c) C( (x/(1+x))/A(x)^2 ) = 1 + x.

(4.d) B( (x/(1+x))/A(x)^2 ) = 1 + x*(1+x)/A(x)^2.

(4.e) A( (x/(1+x))/A(x)^2 + x^2/A(x)^4 ) = 1 + x*(1+x)/A(x)^2.

(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.

(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

(5.c) A145345(n) = [x^(n-1)] C(x)^2 = [x^(n-1)] A(x)^(2*n)/n for n >= 1.