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%I #24 Aug 06 2023 15:36:45
%S 1,0,4,16,90,552,3818,27256,200812,1515912,11695500,91752936,
%T 729850840,5873414168,47739736006,391396504592,3233190109306,
%U 26886605900496,224911668468892,1891436025156672,15982645615936990,135640664826165488,1155711422893896044,9882880183890197352,84794414367362654416,729778564590247004856,6298814833598391313930
%N G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} x^n * A(x)^n * (A(x)^n + x^n)^n.
%C Related identity: Sum_{n=-oo..+oo} x^n * (y - x^n)^n = 0, which holds as a formal power series for all y.
%H Paul D. Hanna, <a href="/A292088/b292088.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A = A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) 2 = Sum_{n=-oo..+oo} (x*A)^n * (A^n + x^n)^n.
%F (2) 2 = Sum_{n=-oo..+oo} (x*A)^(n^2-n) / (A^n + x^n)^n.
%F (3) Sum_{n>=1} (x*A)^(n^2-n)/(A^n + x^n)^n = 2 - Sum_{n>=0} (x*A)^n*(A^n + x^n)^n.
%e G.f.: A(x) = 1 + 4*x^2 + 16*x^3 + 90*x^4 + 552*x^5 + 3818*x^6 + 27256*x^7 + 200812*x^8 + 1515912*x^9 + 11695500*x^10 + 91752936*x^11 + 729850840*x^12 + 5873414168*x^13 + 47739736006*x^14 + 391396504592*x^15 + 3233190109306*x^16 +...
%e Given A = A(x), define
%e P(x) = 1 + (x*A)*(A + x) + (x*A)^2*(A^2 + x^2)^2 + (x*A)^3*(A^3 + x^3)^3 + (x*A)^4*(A^4 + x^4)^4 + (x*A)^5*(A^5 + x^5)^5 +...+ (x*A)^n*(A^n + x^n)^n +...
%e Q(x) = 1/(A + x) + (x*A)^2/(A^2 + x^2)^2 + (x*A)^6/(A^3 + x^3)^3 + (x*A)^12/(A^4 + x^4)^4 + (x*A)^20/(A^5 + x^5)^5 +...+ (x*A)^(n^2-n)/(A^n + x^n)^n +...
%e then P(x) + Q(x) = 2.
%e Explicitly,
%e P(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 357*x^5 + 2411*x^6 + 17101*x^7 + 126047*x^8 + 950172*x^9 + 7324084*x^10 + 57423493*x^11 + 456717652*x^12 + 3675758545*x^13 + 29884252434*x^14 + 245091410895*x^15 + 2025466163355*x^16 + 16851425417853*x^17 + 141038441106711*x^18 + 1186738922293689*x^19 + 10033676061349606*x^20 +...
%e Q(x) = 1 - x - 2*x^2 - 9*x^3 - 63*x^4 - 357*x^5 - 2411*x^6 +...
%e RELATED SERIES.
%e The expansion of the symmetric series
%e S(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (x^n + y^n)^n
%e in terms of x and y begins
%e S(x,y) = (1/y) * ( (y + 1) + (y^4 - 1)*(x/y) + (y^9 + y^4 + y + 1)*(x/y)^2 + (y^16 - 1)*(x/y)^3 + (y^25 + 2*y^9 - 2*y + 1)*(x/y)^4 + (y^36 - 1)*(x/y)^5 + (y^49 + 3*y^16 + y^9 + y^4 + 3*y + 1)*(x/y)^6 + (y^64 - 1)*(x/y)^7 + (y^81 + 4*y^25 - 4*y + 1)*(x/y)^8 + (y^100 + 3*y^16 - 3*y^4 - 1)*(x/y)^9 + (y^121 + 5*y^36 + 5*y + 1)*(x/y)^10 + (y^144 - 1)*(x/y)^11 + (y^169 + 6*y^49 + 6*y^25 + y^16 + y^9 + 6*y^4 - 6*y + 1)*(x/y)^12 + (y^196 - 1)*(x/y)^13 + (y^225 + 7*y^64 + 7*y + 1)*(x/y)^14 + (y^256 + 10*y^36 - 10*y^4 - 1)*(x/y)^15 + (y^289 + 8*y^81 + 4*y^25 - 4*y^9 - 8*y + 1)*(x/y)^16 + ...);
%e substituting y = A(x) in the above simplifies to S(x,y=A(x)) = 2.
%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0); A[#A] = Vec( sum(m=-#A-1,#A+1, x^m*Ser(A)^m * (Ser(A)^m + x^m)^m))[#A]);A[n+1] }
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A361775.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 08 2017
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