A290003 G.f.: A(x) = Sum_{n=-oo..+oo} (x - x^n)^n.

1, -1, 1, -2, 3, -3, 1, 1, 1, -7, 10, -6, 1, 0, 1, -8, 23, -25, 1, 17, 1, -32, 36, -12, 1, -21, 26, -14, 55, -92, 1, 93, 1, -129, 78, -18, 108, -121, 1, -20, 105, -49, 1, 19, 1, -298, 430, -24, 1, -423, 50, 424, 171, -469, 1, -217, 661, -450, 210, -30, 1, -203, 1, -32, 591, -897, 1288, -881, 1, -987, 300, 2407, 1, -2804, 1, -38, 2626, -1350, 1387, -2380, 1, 837, 487, -42, 1, -2855, 3741, -44, 465, -3301, 1, -326, 4291, -2324, 528, -48, 5815, -12713, 1, 6957, 1422, 4074, 1, -10371, 1, -8451, 20322, -54, 1, -15589, 1

Offset

0,4

Comments

Compare g.f. to: Sum_{n=-oo..+oo} (x - x^(n+1))^n = 0.

Links

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

Formula

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

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

From Peter Bala, Mar 02 2025: (Start)

The above conjecture follows from the following formula: for n >= 2,

a(n) = Sum_{d divides n-1} (-1)^(d-1) * ( binomial((n-1)/d, d-2) + binomial((n+d-1)/d, d-1) ).

For prime p >= 3, a(p^2 + 1) = p^2 + 1. (End)

Example

G.f.: A(x) = 1 - x + x^2 - 2*x^3 + 3*x^4 - 3*x^5 + x^6 + x^7 + x^8 - 7*x^9 + 10*x^10 - 6*x^11 + x^12 + x^14 - 8*x^15 + 23*x^16 - 25*x^17 + x^18 + 17*x^19 + x^20 - 32*x^21 + 36*x^22 - 12*x^23 + x^24 - 21*x^25 + 26*x^26 - 14*x^27 + 55*x^28 - 92*x^29 + x^30 +...
where A(x) = 1 + P(x) + N(x) with
P(x) = (x-x) + (x-x^2)^2 + (x-x^3)^3 + (x-x^4)^4 + (x-x^5)^5 + (x-x^6)^6 + (x-x^7)^7 +...+ (x-x^n)^n +...
N(x) = -x/(1 - x^2) + x^4/(1-x^3)^2 - x^9/(1-x^4)^3 + x^16/(1-x^5)^4 - x^25/(1-x^6)^5 +...+ (-x)^(n^2)/(1-x^(n+1))^n +...
Explicitly,
P(x) = x^2 - x^3 + 2*x^4 - 2*x^5 + x^6 + x^8 - 5*x^9 + 7*x^10 - 5*x^11 + x^12 + x^14 - 7*x^15 + 17*x^16 - 18*x^17 + x^18 + 12*x^19 + x^20 - 25*x^21 + 29*x^22 - 11*x^23 + x^24 - 12*x^25 + 16*x^26 - 13*x^27 + 46*x^28 - 70*x^29 + x^30 +...
N(x) = -x - x^3 + x^4 - x^5 + x^7 - 2*x^9 + 3*x^10 - x^11 - x^15 + 6*x^16 - 7*x^17 + 5*x^19 - 7*x^21 + 7*x^22 - x^23 - 9*x^25 + 10*x^26 - x^27 + 9*x^28 - 22*x^29 +...
From Paul D. Hanna, Jan 13 2025: (Start)
SPECIAL VALUES.
A local maximum of A(x) is at x = z, A'(z) = 0,
  where z = 0.6783626505745664596168958924200373689742586374321477329...
  and A(z) = 0.332320805615430858829730480236535256165083297416146964...
A(5/6) = 0.30801526795391347776371668529063511729774504098314...
A(4/5) = 0.31797024517441016604092565708098992009134940089362...
A(3/4) = 0.32759707660987407896902126812991555844980484348844...
A(2/3) = 0.33220302782561874934924055409715505666564907222676...
A(3/5) = 0.32724657183605678719721082848286562112862495149949...
A(1/2) = 0.30725396830704316799197832656390411971168116373389...
A(2/5) = 0.27337943400586708871078028747061201307317280175586...
A(1/3) = 0.24338606674563424484910361835257533242309621632065...
A(1/4) = 0.19758524006807690544490179709803177425355852401229...
A(1/5) = 0.16558333624735433324843855679493132539350188690309...
A(1/6) = 0.14230098666491512550971306545368484826875874989347...
(End)

Maple

with(numtheory):
seq(add((-1)^(d-1) * (binomial((n-1)/d, d-2) + binomial((n+d-1)/d, d-1)), d in divisors(n-1)), n = 2..110); # Peter Bala, Mar 02 2025

Prog

(PARI) {a(n) = local(A=1); A = sum(k=-n, n, (x - x^k)^k +x*O(x^n)); polcoeff(A, n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = 1 + sum(k=1, n, x^k*(1 - x^(k-1))^k + (-x)^(k^2)/(1 - x^(k+1))^k +x*O(x^n)); polcoeff(A, n)}
for(n=0, 500, print1(a(n), ", "))

Crossrefs

Cf. A260116, A260147, A378582, A379195.

Sequence in context: A318741 A171872 A005135 * A139460 A105244 A257451

Adjacent sequences: A290000 A290001 A290002 * A290004 A290005 A290006

Keyword

sign,easy,changed

Author

Paul D. Hanna, Sep 03 2017

Status

approved