A260147 - OEIS

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A260147

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

%I #71 Aug 02 2023 20:11:30

%S 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,

%T 236,287,307,30,30,1141,32,1,727,681,1247,314,38,970,1652,1821,42,

%U 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

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

%C Compare to the curious identities:

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

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

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

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

%H Paul D. Hanna, <a href="/A260147/b260147.txt">Table of n, a(n) for n = 0..2050</a>

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

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

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

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

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

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

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

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

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

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

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

%F From _Peter Bala_, Jan 23 2021: (Start)

%F The following are conjectural:

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

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

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

%F 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)).

%F 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).

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

%F 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).

%F 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).

%F 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).

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

%F 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)).

%F (End)

%e 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 +...

%e where 2*A(x) = 1 + P(x) + N(x) with

%e 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 +...

%e 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 +...

%e Explicitly,

%e 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 +...

%e 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 +...

%t 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 *)

%o (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)}