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

1, 1, 3, 14, 85, 615, 5038, 45265, 437012, 4472197, 48056889, 538621852, 6265669760, 75369364118, 934809950418, 11928201381716, 156302591148741, 2100191239445909, 28901831807930949, 406933300084065353, 5857010329019250612, 86111062850900773745, 1292373792900901543026, 19788451519046405896069

Offset

1,3

Links

Table of n, a(n) for n=1..24.

Formula

G.f. B(x) and A(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.: 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 +...
such that A(x - A(x)*B(x)) = x and B(x - x*A(x)) = x where
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 +...

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(B, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A263186.

Sequence in context: A230218 A301934 A160881 * A213628 A088716 A005189

Adjacent sequences: A263184 A263185 A263186 * A263188 A263189 A263190

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2015

Status

approved