cp's OEIS Frontend

A302704 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx.

%I A302704 #9 Apr 20 2018 00:00:53
%S A302704 1,1,3,7,10,24,186,492,-1863,-5240,79369,220350,-2492912,-6984296,
%T A302704 90693060,254955852,-3412605726,-9625060440,133881917577,378533393025,
%U A302704 -5412043255536,-15332556581976,224289105628470,636469447338144,-9487486533101850,-26960087538403992,408305313050817591,1161625141535962012,-17832202665017550896,-50783861201670203640,788741951929695672520
%N A302704 O.g.f. A(x) satisfies:  A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx.
%H A302704 Paul D. Hanna, <a href="/A302704/b302704.txt">Table of n, a(n) for n = 0..400</a>
%F A302704 O.g.f. A(x) satisfies:
%F A302704 (1) A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx.
%F A302704 (2) A(x) = 1 + Integral A(x)^3 * (A(x) + 8*x*A'(x)) / (A(x) + 5*x*A'(x)) dx.
%F A302704 (3) A(x) = 1 + Integral A(x) * (sqrt( 1 + 4*x*A(x)^2 + 64*x^2*A(x)^4 ) - (1 - 8*x*A(x)^2))/(10*x) dx.
%F A302704 (4) 0 = A(x)^4 - A(x)*(1 - 8*x*A(x)^2)*A'(x) - 5*x*A'(x)^2.
%e A302704 G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 10*x^4 + 24*x^5 + 186*x^6 + 492*x^7 - 1863*x^8 - 5240*x^9 + 79369*x^10 + 220350*x^11 - 2492912*x^12 + ...
%e A302704 RELATED SERIES.
%e A302704 (x*A(x)^8)' / (x*A(x)^5)' = 1 + 6*x + 21*x^2 + 40*x^3 + 120*x^4 + 1116*x^5 + 3444*x^6 - 14904*x^7 - 47160*x^8 + 793690*x^9 + 2423850*x^10 + ...
%e A302704 which equals A'(x).
%e A302704 The logarithmic derivative of the g.f. begins:
%e A302704 A'(x)/A(x) = 1 + 5*x + 13*x^2 + 5*x^3 + 31*x^4 + 905*x^5 + 1975*x^6 - 21595*x^7 - 41270*x^8 + 883355*x^9 + 1736824*x^10 - 34567735*x^11 + ...
%e A302704 which equals (sqrt(1 + 4*x*A(x)^2 + 64*x^2*A(x)^4) - (1 - 8*x*A(x)^2))/(10*x).
%o A302704 (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x*A^8)'/(x*A^5 +x*O(x^n))' ); ); polcoeff(A, n)}
%o A302704 for(n=0,30,print1(a(n),", "))
%Y A302704 Cf. A302701, A302705, A303064.
%K A302704 sign
%O A302704 0,3
%A A302704 _Paul D. Hanna_, Apr 19 2018