A387041 G.f. A(x) satisfies (A(x) - x^2) o (x - A(x)^2) = x, where operator 'o' denotes composition.

1, 2, 6, 41, 348, 3360, 35632, 406104, 4904914, 62180918, 821752456, 11263836924, 159523476148, 2327336091732, 34894961587312, 536671299862721, 8453184479505430, 136188177741639378, 2241801065131393700, 37670062720274627960, 645649822816127973456, 11279877783091509190416

Offset

1,2

Links

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = x + x^2 + A(A(x) - x^2)^2.

(2) A(x - A(x)^2) = x + (x - A(x)^2)^2.

(3) A(A(x) - x^2) = sqrt( A(x) - x^2 - x ).

(4) A(x)^2 = x - sqrt( A(x - A(x)^2) - x ).

(5) A(x) = x^2 + Series_Reversion(x - A(x)^2).

(6) A(x) = sqrt( x - Series_Reversion(A(x) - x^2) ).

(7) A(x) = x + x^2 + Sum_{n>=0} d^n/dx^n A(x)^(2*n+2) / (n+1)!.

(8) A(x) = x^2 + x*exp( Sum_{n>=0} d^n/dx^n (A(x)^(2*n+2)/x) / (n+1)! ).

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 41*x^4 + 348*x^5 + 3360*x^6 + 35632*x^7 + 406104*x^8 + 4904914*x^9 + 62180918*x^10 + ...
where A(x) - x^2 = x + A(A(x) - x^2)^2;
also, A(x - A(x)^2) = x + (x - A(x)^2)^2 = x + x^2 - 2*x*A(x)^2 + A(x)^4.
RELATED SERIES.
A(A(x) - x^2) = x + 3*x^2 + 16*x^3 + 126*x^4 + 1174*x^5 + 12278*x^6 + 139496*x^7 + 1689597*x^8 + 21553566*x^9 + 287191110*x^10 + ...
A(x - A(x)^2) = x + x^2 - 2*x^3 - 7*x^4 - 24*x^5 - 164*x^6 - 1452*x^7 - 14312*x^8 - 153354*x^9 - 1757322*x^10 - ...
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 106*x^5 + 896*x^6 + 8604*x^7 + 90561*x^8 + 1023592*x^9 + 12258452*x^10 + ...
A(x)^4 = x^4 + 8*x^5 + 48*x^6 + 340*x^7 + 2896*x^8 + 27768*x^9 + 289862*x^10 + ...

Prog

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

Crossrefs

Sequence in context: A367603 A381896 A336281 * A326268 A096138 A004153

Adjacent sequences: A387038 A387039 A387040 * A387042 A387043 A387044

Keyword

nonn

Author

Paul D. Hanna, Aug 14 2025

Status

approved