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%I #12 May 11 2019 21:56:40
%S 1,9,26,224,2476,23224,287376,3757374,52257501,788478999,12610776876,
%T 212979510624,3790898971876,70777961166874,1381742116784376,
%U 28138911700128124,596359893046340626,13125567196055049999,299467375895770156251,7070998303069778171874,172523963169549496984376,4343634679080455046328124,112703773700255237721093751,3010180318995682243232265624,82668711264219381166762578126
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 5*x)^n - A(x))^(n+1), where A(0) = 0.
%H Paul D. Hanna, <a href="/A325585/b325585.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1 + 5*x)^n - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1 + 5*x)^(n*(n-1)) / (1 + x*(1 + 5*x)^n*A(x))^(n+1).
%F FORMULA FOR TERMS.
%F a(n) = (-1)^n (mod 5) for n >= 0.
%e G.f.: A(x) = x + 9*x^2 + 26*x^3 + 224*x^4 + 2476*x^5 + 23224*x^6 + 287376*x^7 + 3757374*x^8 + 52257501*x^9 + 788478999*x^10 + 12610776876*x^11 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+5*x) - A(x))^2 + x^2*((1+5*x)^2 - A(x))^3 + x^3*((1+5*x)^3 - A(x))^4 + x^4*((1+5*x)^4 - A(x))^5 + x^5*((1+5*x)^5 - A(x))^6 + x^6*((1+5*x)^6 - A(x))^7 + ...
%t a[n_] := Module[{A}, A={1}; For[i=1, i <= n, i++, AppendTo[A, 0]; A[[-1]] = Coefficient[Sum[x^m*((1 + 5*x + x*O[x]^Length[A])^m - x*(A.x^Range[0, Length[A]-1]))^(m+1), {m, 0, Length[A]}], x, Length[A]]]; A[[n+1]]];
%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, May 11 2019, from PARI *)
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1 + 5*x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A325582, A325583, A325584.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 11 2019
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