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A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.

%I A321604 #9 Oct 25 2020 13:38:35
%S A321604 1,4,22,564,25157,1499576,109904860,9480509576,937113401201,
%T A321604 104240673195936,12878161899791760,1749261564410844864,
%U A321604 259132759251207789056,41585481940418457992816,7188476201158569394613976,1331880173688346226092103696,263358773243148578509342224153,55363099822436514905885084770968,12330972024423209530808891225876436,2900976547500300324930009436969260936
%N A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
%H A321604 Paul D. Hanna, <a href="/A321604/b321604.txt">Table of n, a(n) for n = 0..200</a>
%F A321604 G.f. A(x) satisfies:
%F A321604 (1) 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
%F A321604 (2) 1 = Sum_{n>=0} (1+x)^(4*n^2) / (1 + (1+x)^(4*n)*A(x))^(n+1).
%e A321604 G.f.: A(x) = 1 + 4*x + 22*x^2 + 564*x^3 + 25157*x^4 + 1499576*x^5 + 109904860*x^6 + 9480509576*x^7 + 937113401201*x^8 + 104240673195936*x^9 + ...
%e A321604 such that
%e A321604 1 = 1  +  ((1+x)^4 - A(x))  +  ((1+x)^8 - A(x))^2  +  ((1+x)^12 - A(x))^3  +  ((1+x)^16 - A(x))^4  +  ((1+x)^20 - A(x))^5  +  ((1+x)^24 - A(x))^6  +  ((1+x)^28 - A(x))^7 + ...
%e A321604 Also,
%e A321604 1 = 1/(1 + A(x))  +  (1+x)^4/(1 + (1+x)^4*A(x))^2  +  (1+x)^16/(1 + (1+x)^8*A(x))^3  +  (1+x)^36/(1 + (1+x)^12*A(x))^4  +  (1+x)^64/(1 + (1+x)^16*A(x))^5  +  (1+x)^100/(1 + (1+x)^20*A(x))^6  + ...
%e A321604 RELATED SERIES.
%e A321604 The logarithmic derivative of the g.f. begins
%e A321604 A'(x)/A(x) = 4 + 28*x + 1492*x^2 + 91788*x^3 + 6981484*x^4 + 621939700*x^5 + 63151305340*x^6 + 7181135905380*x^7 + 903210250234696*x^8 + ...
%e A321604 the coefficients of which are all divisible by 4:
%e A321604 (1/4) * A'(x)/A(x) = 1 + 7*x + 373*x^2 + 22947*x^3 + 1745371*x^4 + 155484925*x^5 + 15787826335*x^6 + 1795283976345*x^7 + 225802562558674*x^8 + ...
%o A321604 (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(4*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
%o A321604 for(n=0,30,print1(a(n),", "))
%Y A321604 Cf. A303056, A321602, A321603, A321605.
%Y A321604 Cf. A326264.
%K A321604 nonn
%O A321604 0,2
%A A321604 _Paul D. Hanna_, Nov 14 2018