A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

2, 4, 2, 10, 2, 12, 16, 16, 2, 50, 24, 24, 58, 28, 72, 154, 2, 36, 302, 40, 142, 270, 332, 48, 242, 472, 574, 614, 60, 60, 2282, 64, 2, 1454, 1362, 2494, 628, 76, 1940, 3304, 3642, 84, 5266, 88, 662, 13180, 3544, 96, 994, 6106, 14292, 13602, 3434, 108, 8102, 14854, 16018, 24778, 7310, 120, 35684

Offset

0,1

Comments

Compare to the curious identities:

(1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

(2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n = 0.

Links

Table of n, a(n) for n=0..60.

Formula

G.f.: Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.

G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.

G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.

G.f.: Sum_{n>=1} x^(n^2-n) *((1 + x^n)^n + (1 - x^n)^n) / (1 - x^(2*n))^n.

G.f.: Sum_{n=-oo..+oo} x^n * ((1 + x^n)^(2*n) + (1 - x^n)^(2*n)) = Sum_{n>=0} a(n)*x^n.

a(n) = 2*A260147(n).

a(2^n) = 2 for n > 0 (conjecture).

a(p) = 2*p+2 for primes p > 3 (conjecture).

Example

G.f.: A(x) = 2 + 4*x^2 + 2*x^4 + 10*x^6 + 2*x^8 + 12*x^10 + 16*x^12 + 16*x^14 + 2*x^16 + 50*x^18 + 24*x^20 +...
where A(x) = 1 + P(x) + N(x) with
P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 +...
N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 +...
Explicitly,
P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n +...
N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n +...

Mathematica

terms = 100; max = 2 terms; Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *)

Prog

(PARI) {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k  + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = sum(k=1, sqrtint(2*n)+2, x^(k^2-k) *((1 + x^k)^k + (1 - x^k)^k) / (1 - x^(2*k)  + O(x^(2*n+2)) )^k ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k)) + O(x^(n+1)) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561.

Sequence in context: A353108 A072866 A061393 * A055935 A086930 A235798

Adjacent sequences: A260358 A260359 A260360 * A260362 A260363 A260364

Keyword

nonn

Author

Paul D. Hanna, Jul 23 2015

Status

approved