A263186 G.f. A(x) satisfies: A( x - A(x)*B(x) ) = x such that B( x - x*A(x) ) = x, where B(x) is the g.f. of A263187.

1, 1, 4, 23, 160, 1260, 10861, 100474, 984944, 10142888, 109039530, 1218011097, 14086708075, 168205533546, 2069043383080, 26170130114863, 339856785957307, 4525776548471074, 61735297113781725, 861823700018556599, 12302696382913051859, 179461986070686773966, 2673380518707453159859

Offset

1,3

Links

Table of n, a(n) for n=1..23.

Formula

G.f. A(x) and B(x) satisfy the differential series:

(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * B(x)^n / n!.

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

(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * B(x)^n / (n!*x) ).

(4) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A(x)^n / n! ).

Example

 G.f.: A(x) = x + x^2 + 4*x^3 + 23*x^4 + 160*x^5 + 1260*x^6 + 10861*x^7 + 100474*x^8 + 984944*x^9 + 10142888*x^10 + 109039530*x^11 +...
such that A(x - A(x)*B(x)) = x and B(x - x*A(x)) = x where
B(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 615*x^6 + 5038*x^7 + 45265*x^8 + 437012*x^9 + 4472197*x^10 + 48056889*x^11 +...

Prog

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

Crossrefs

Cf. A263187.

Sequence in context: A326350 A198916 A182969 * A245110 A342988 A304074

Adjacent sequences: A263183 A263184 A263185 * A263187 A263188 A263189

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2015

Status

approved