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A321605 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.

%I A321605 #9 Oct 25 2020 13:39:49
%S A321605 1,5,35,1110,61830,4607001,422112085,45521033720,5625206604320,
%T A321605 782244114339935,120812011501389376,20514224767917807795,
%U A321605 3798925417133114909240,762102329400356260363990,164678708686403817727101920,38140958485665617437764886383,9427520984195812306085385378080,2477372683628569966077893189614835,689743886246438120027048924784220410
%N A321605 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
%H A321605 Paul D. Hanna, <a href="/A321605/b321605.txt">Table of n, a(n) for n = 0..200</a>
%F A321605 G.f. A(x) satisfies:
%F A321605 (1) 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
%F A321605 (2) 1 = Sum_{n>=0} (1+x)^(5*n^2) / (1 + (1+x)^(5*n)*A(x))^(n+1).
%e A321605 G.f.: A(x) = 1 + 5*x + 35*x^2 + 1110*x^3 + 61830*x^4 + 4607001*x^5 + 422112085*x^6 + 45521033720*x^7 + 5625206604320*x^8 + 782244114339935*x^9 + ...
%e A321605 such that
%e A321605 1 = 1  +  ((1+x)^5 - A(x))  +  ((1+x)^10 - A(x))^2  +  ((1+x)^15 - A(x))^3  +  ((1+x)^20 - A(x))^4  +  ((1+x)^25 - A(x))^5  +  ((1+x)^30 - A(x))^6  +  ((1+x)^35 - A(x))^7 + ...
%e A321605 Also,
%e A321605 1 = 1/(1 + A(x))  +  (1+x)^5/(1 + (1+x)^5*A(x))^2  +  (1+x)^20/(1 + (1+x)^10*A(x))^3  +  (1+x)^45/(1 + (1+x)^15*A(x))^4  +  (1+x)^80/(1 + (1+x)^20*A(x))^5  +  (1+x)^125/(1 + (1+x)^25*A(x))^6  + ...
%e A321605 RELATED SERIES.
%e A321605 The logarithmic derivative of the g.f. begins
%e A321605 A'(x)/A(x) = 5 + 45*x + 2930*x^2 + 225545*x^3 + 21445630*x^4 + 2388480630*x^5 + 303204843520*x^6 + 43104182972905*x^7 + 6777636393880895*x^8 + ...
%e A321605 the coefficients of which are all divisible by 5:
%e A321605 (1/5) * A'(x)/A(x) = 1 + 9*x + 586*x^2 + 45109*x^3 + 4289126*x^4 + 477696126*x^5 + 60640968704*x^6 + 8620836594581*x^7 + 1355527278776179*x^8 + ...
%o A321605 (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(5*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
%o A321605 for(n=0,30,print1(a(n),", "))
%Y A321605 Cf. A303056, A321602, A321603, A321604.
%Y A321605 Cf. A326265.
%K A321605 nonn
%O A321605 0,2
%A A321605 _Paul D. Hanna_, Nov 14 2018