A260147 G.f.: (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

1, 2, 1, 5, 1, 6, 8, 8, 1, 25, 12, 12, 29, 14, 36, 77, 1, 18, 151, 20, 71, 135, 166, 24, 121, 236, 287, 307, 30, 30, 1141, 32, 1, 727, 681, 1247, 314, 38, 970, 1652, 1821, 42, 2633, 44, 331, 6590, 1772, 48, 497, 3053, 7146, 6801, 1717, 54, 4051, 7427, 8009, 12389, 3655, 60, 17842, 62, 4496, 42841, 1, 15731, 6470, 68, 19449, 34754, 65781

Offset

0,2

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.

Given G(x,q) = Sum_{n=-oo..+oo} (1 + q^n)^n * q^n * x^n, then

[x^0] G(x,q)^2 = theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Formula

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

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

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

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

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

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

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

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

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

a(2^n) = 1 for n > 0 (conjecture).

a(p) = p+1 for primes p > 3 (conjecture).

From Peter Bala, Jan 23 2021: (Start)

The following are conjectural:

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

Equivalently: A(x^2) = Sum_{n = -oo..+oo} x^(4*n^2 + 2*n)/(1 + x^(2*n+1))^(2*n+1).

a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(4*n+2)

More generally, for k = 1,2,3,..., a((2^k)*(2*n + 1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(2^(k+1)*(2*n+1)).

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

More generally, for k = 1,2,3,...,

a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2^(k+1)*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2^(k+1)*n).

a(4*n+2) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 + x^n)^(2*n) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 - x^n)^(2*n).

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

For k = 1,2,3,...,

a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2^(k+1)*(2*n+1)).

(End)

Example

G.f.: A(x) = 1 + 2*x^2 + x^4 + 5*x^6 + x^8 + 6*x^10 + 8*x^12 + 8*x^14 + x^16 + 25*x^18 + 12*x^20 +...
where 2*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; 1/2 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/2 + 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 /2  + 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=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k))/2 + O(x^(n+1)) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = sum(k=-n-1, n+1, x^k*(1+x^k)^(2*k) + O(x^(n+1)) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = sum(k=-n-1, n+1, x^(2*k^2-k)/(1-x^k + O(x^(n+1)))^(2*k)  ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A260116, A260148, A217668, A260180, A260361.

Cf. A261605, A363558, A363559, A363569, A363561.

Sequence in context: A249548 A014650 A014648 * A263454 A036073 A124227

Adjacent sequences: A260144 A260145 A260146 * A260148 A260149 A260150

Keyword

nonn,easy

Author

Paul D. Hanna, Jul 17 2015

Status

approved