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.

2, 1, 1, -3, 7, 15, -39, -307, 917, 2540, -16939, -25016, 441962, -498346, -10210949, 42714405, 195220459, -2142879945, 532985665, 83535107090, -365902332521, -2233273290797, 28143121253695, -20874136499710, -1436795595314700, 8862053852144592, 38496064560804831

Offset: 0

Author

Paul D. Hanna, Sep 04 2022

Keywords

• sign

Comments

By construction, the symmetric generating function A(x) formally obeys the functional equation A(1-x) = A(x).

Examples

			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 + ...
such that A(x) = P(x) + N(x)
where
P(x) = Sum_{n>=0} (x - x^2)^n * ((1-x)^n + x^n)^n,
N(x) = Sum_{n>=1} (x - x^2)^(n*(n-1)) / ((1-x)^n + x^n)^n ;
explicitly,
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 + ...
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 + ...
Formally, P(1/2) = N(1/2) = A(1/2) / 2 = Sum_{n>=0} 1/2^(n*(n+1)) = A319016.
SPECIFIC VALUES.
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.
A(1/2) = 2 * Sum_{n>=0} 1 / 2^(n*(n+1)) = 2 * A319016.
A(1/3) = 3 * Sum_{n=-oo..+oo} 2^n * (2^n + 1)^n / 3^((n+1)^2).
A(1/4) = 4 * Sum_{n=-oo..+oo} 3^n * (3^n + 1)^n / 4^((n+1)^2).
A(1/5) = 5 * Sum_{n=-oo..+oo} 4^n * (4^n + 1)^n / 5^((n+1)^2).
...
Explicitly,
A(1/2) = 2.53174019046173273683784262908708685492853089279927743364010...
A(1/3) = A(2/3) = 2.40113401345619748217278781385876768642497840209318...
A(1/4) = A(3/4) = 2.29231716604599371012382686555008867061019357832301...
A(1/5) = A(4/5) = 2.22791516133132211947724853770151015273548188586981...
...
A(0) = A(1) = 2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1050

Crossrefs

Cf. A319016.

Programs

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)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x) = A(1 - x).

(2) A(x) = Sum_{n=-oo..+oo} (x - x^2)^n * ((1-x)^n + x^n)^n.

(3) A(x) = Sum_{n=-oo..+oo} (x - x^2)^(n*(n-1)) / ((1-x)^n + x^n)^n.