A263186 - OEIS

%I #7 Sep 08 2024 00:03:58

%S 1,1,4,23,160,1260,10861,100474,984944,10142888,109039530,1218011097,

%T 14086708075,168205533546,2069043383080,26170130114863,

%U 339856785957307,4525776548471074,61735297113781725,861823700018556599,12302696382913051859,179461986070686773966,2673380518707453159859

%N 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.

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

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

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

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

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

%e 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 +...

%e such that A(x - A(x)*B(x)) = x and B(x - x*A(x)) = x where

%e 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 +...

%o (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)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A263187.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Nov 03 2015