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%I #3 Nov 03 2015 18:32:06
%S 1,1,3,14,85,615,5038,45265,437012,4472197,48056889,538621852,
%T 6265669760,75369364118,934809950418,11928201381716,156302591148741,
%U 2100191239445909,28901831807930949,406933300084065353,5857010329019250612,86111062850900773745,1292373792900901543026,19788451519046405896069
%N 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.
%F G.f. B(x) and A(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.: 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 +...
%e such that A(x - A(x)*B(x)) = x and B(x - x*A(x)) = x where
%e 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 +...
%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(B,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A263186.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Nov 03 2015
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