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A358962 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).

%I A358962 #21 Dec 09 2022 14:33:22
%S A358962 1,2,8,30,146,748,4002,22114,125220,722850,4238148,25169064,151084168,
%T A358962 915235106,5587985801,34351213384,212436911849,1320744403708,
%U A358962 8250065775120,51752790871466,325887027304769,2059216160242430,13052805881695018,82976612756731258
%N A358962 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).
%C A358962 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(3*n+2))^n, which holds formally for all y.
%H A358962 Paul D. Hanna, <a href="/A358962/b358962.txt">Table of n, a(n) for n = 0..200</a>
%F A358962 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A358962 (1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).
%F A358962 (2) x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n^2) / (1 - x^(3*n-2)*A(x))^(n+1).
%F A358962 (3) A(x) = Sum_{n=-oo..+oo} x^(4*n+2)* (A(x) - x^(3*n+2))^(n-1).
%F A358962 (4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n*(n-1)) / (1 - x^(3*n-2)*A(x))^(n+1).
%F A358962 (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n*(n-1)) / (1 - x^(3*n-2)*A(x))^n.
%e A358962 G.f.: A(x) =  1 + 2*x + 8*x^2 + 30*x^3 + 146*x^4 + 748*x^5 + 4002*x^6 + 22114*x^7 + 125220*x^8 + 722850*x^9 + 4238148*x^10 + ...
%e A358962 where A = A(x) satisfies the doubly infinite sum
%e A358962 1 = ... + x^(-2)*(A - x^(-4))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^2)^(-1) + x*(A - x^5)^0 + x^2*(A - x^8) + x^3*(A - x^11)^2 + x^4*(A - x^14)^3 + ... + x^n * (A - x^(3*n+2))^(n-1) + ...
%e A358962 also,
%e A358962 A(x) = ... + x^36/(1 - x^(-11)*A)^(-2) - x^18/(1 - x^(-8)*A)^(-1) + x^6 - 1/(1 - x^(-2)*A) + 1/(1 - x*A)^2 - x^6/(1 - x^4*A)^3 + x^18/(1 - x^7*A)^4 - x^36/(1 - x^10*A)^5 + ... + (-1)^(n+1)*x^(3*n*(n-1))/(1 - x^(3*n-2)*A)^(n+1) + ...
%o A358962 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A358962 A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(3*n+2))^(n-1) ), #A-1) );A[n+1]}
%o A358962 for(n=0,30,print1(a(n),", "))
%Y A358962 Cf. A358961, A358963, A358964, A358965.
%K A358962 nonn
%O A358962 0,2
%A A358962 _Paul D. Hanna_, Dec 07 2022