A258936 G.f.: Sum_{n=-oo..+oo} x^n * (1 - 2^n*x^n)^n.

-1, -3, 7, -15, 89, -63, 121, -255, 3521, -13119, 18273, -4095, -40319, -16383, 425089, -2676735, 6141953, -262143, -22487551, -1048575, 173791233, -356171775, 176138241, -16777215, 2378907649, -5430575103, 3355336705, -38913703935, 164745740289, -1073741823, -770681831423, -4294967295, 4113638096897, -3796520402943, 1133869137921, -38542231207935, 87257121292289, -274877906943

Offset: 0

Author

Paul D. Hanna, Nov 06 2015

Keywords

• sign

Comments

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

More generally, for all k we have the identity:

Sum_{n=-oo..+oo} x^n * (1 - k^n*x^n)^n = (-1) * Sum_{n=-oo..+oo} k*(k*x)^n * (1 - k*(k*x)^n)^n. - Paul D. Hanna, Dec 25 2015

Examples

			G.f.: A(x) = -1 - 3*x + 7*x^2 - 15*x^3 + 89*x^4 - 63*x^5 + 121*x^6 - 255*x^7 + 3521*x^8 - 13119*x^9 + 18273*x^10 - 4095*x^11 - 40319*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A260147.

Programs

PARI

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

Formula

G.f.: (-1) * Sum_{n=-oo..+oo} 2*(2*x)^n * (1 - 2*(2*x)^n)^n. - Paul D. Hanna, Dec 25 2015

It appears that for prime p >= 3, a(p) = 1 - 2^(p+1). - Peter Bala, Aug 06 2023