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A356776 a(n) = coefficient in the power series expansion of A(x) = Sum_{n=-oo..+oo} x^n * (1-x)^n * ((1-x)^n + x^n)^n.

%I A356776 #11 Dec 03 2022 12:05:42
%S A356776 2,1,1,-3,7,15,-39,-307,917,2540,-16939,-25016,441962,-498346,
%T A356776 -10210949,42714405,195220459,-2142879945,532985665,83535107090,
%U A356776 -365902332521,-2233273290797,28143121253695,-20874136499710,-1436795595314700,8862053852144592,38496064560804831
%N A356776 a(n) = coefficient in the power series expansion of A(x) = Sum_{n=-oo..+oo} x^n * (1-x)^n * ((1-x)^n + x^n)^n.
%C A356776 By construction, the symmetric generating function A(x) formally obeys the functional equation A(1-x) = A(x).
%H A356776 Paul D. Hanna, <a href="/A356776/b356776.txt">Table of n, a(n) for n = 0..1050</a>
%F A356776 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A356776 (1) A(x) = A(1 - x).
%F A356776 (2) A(x) = Sum_{n=-oo..+oo} (x - x^2)^n * ((1-x)^n + x^n)^n.
%F A356776 (3) A(x) = Sum_{n=-oo..+oo} (x - x^2)^(n*(n-1)) / ((1-x)^n + x^n)^n.
%e A356776 G.f. A(x) = 2 + x + x^2 - 3*x^3 + 7*x^4 + 15*x^5 - 39*x^6 - 307*x^7 + 917*x^8 + 2540*x^9 - 16939*x^10 - 25016*x^11 + 441962*x^12 + ...
%e A356776 such that A(x) = P(x) + N(x)
%e A356776 where
%e A356776 P(x) = Sum_{n>=0} (x - x^2)^n * ((1-x)^n + x^n)^n,
%e A356776 N(x) = Sum_{n>=1} (x - x^2)^(n*(n-1)) / ((1-x)^n + x^n)^n ;
%e A356776 explicitly,
%e A356776 P(x) = 1 + x - 5*x^3 + 6*x^4 + 19*x^5 - 28*x^6 - 294*x^7 + 915*x^8 + 2501*x^9 - 17016*x^10 - 25070*x^11 + 442079*x^12 - 497870*x^13 - 10210070*x^14 + 42715248*x^15 + ...
%e A356776 N(x) = 1 + x^2 + 2*x^3 + x^4 - 4*x^5 - 11*x^6 - 13*x^7 + 2*x^8 + 39*x^9 + 77*x^10 + 54*x^11 - 117*x^12 - 476*x^13 - 879*x^14 - 843*x^15 + 507*x^16 + ...
%e A356776 Formally, P(1/2) = N(1/2) = A(1/2) / 2 = Sum_{n>=0} 1/2^(n*(n+1)) = A319016.
%e A356776 SPECIFIC VALUES.
%e A356776 Even though A(x) as the sum of a power series in x may diverge, the function A(x) may be formally evaluated at each x given below to have the respective specific value.
%e A356776 A(1/2) = 2 * Sum_{n>=0} 1 / 2^(n*(n+1)) = 2 * A319016.
%e A356776 A(1/3) = 3 * Sum_{n=-oo..+oo} 2^n * (2^n + 1)^n / 3^((n+1)^2).
%e A356776 A(1/4) = 4 * Sum_{n=-oo..+oo} 3^n * (3^n + 1)^n / 4^((n+1)^2).
%e A356776 A(1/5) = 5 * Sum_{n=-oo..+oo} 4^n * (4^n + 1)^n / 5^((n+1)^2).
%e A356776 ...
%e A356776 Explicitly,
%e A356776 A(1/2) = 2.53174019046173273683784262908708685492853089279927743364010...
%e A356776 A(1/3) = A(2/3) = 2.40113401345619748217278781385876768642497840209318...
%e A356776 A(1/4) = A(3/4) = 2.29231716604599371012382686555008867061019357832301...
%e A356776 A(1/5) = A(4/5) = 2.22791516133132211947724853770151015273548188586981...
%e A356776 ...
%e A356776 A(0) = A(1) = 2.
%o A356776 (PARI) {a(n) = my(A = sum(m=-n-1,n+1, x^m  * (1-x)^m * ((1-x)^m + x^m +x*O(x^n))^m ) ); polcoeff(A,n)}
%o A356776 for(n=0,30,print1(a(n),", "))
%Y A356776 Cf. A319016.
%K A356776 sign
%O A356776 0,1
%A A356776 _Paul D. Hanna_, Sep 04 2022