A290003 - OEIS

%I #21 Mar 03 2025 13:32:05

%S 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,

%T -21,26,-14,55,-92,1,93,1,-129,78,-18,108,-121,1,-20,105,-49,1,19,1,

%U -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

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

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

%H Paul D. Hanna, <a href="/A290003/b290003.txt">Table of n, a(n) for n = 0..2050</a>

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

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

%F From _Peter Bala_, Mar 02 2025: (Start)

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

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

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

%e 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 +...

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

%e 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 +...

%e 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 +...

%e Explicitly,

%e 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 +...

%e 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 +...

%e From _Paul D. Hanna_, Jan 13 2025: (Start)

%e SPECIAL VALUES.

%e A local maximum of A(x) is at x = z, A'(z) = 0,

%e   where z = 0.6783626505745664596168958924200373689742586374321477329...

%e   and A(z) = 0.332320805615430858829730480236535256165083297416146964...

%e A(5/6) = 0.30801526795391347776371668529063511729774504098314...

%e A(4/5) = 0.31797024517441016604092565708098992009134940089362...

%e A(3/4) = 0.32759707660987407896902126812991555844980484348844...

%e A(2/3) = 0.33220302782561874934924055409715505666564907222676...

%e A(3/5) = 0.32724657183605678719721082848286562112862495149949...

%e A(1/2) = 0.30725396830704316799197832656390411971168116373389...

%e A(2/5) = 0.27337943400586708871078028747061201307317280175586...

%e A(1/3) = 0.24338606674563424484910361835257533242309621632065...

%e A(1/4) = 0.19758524006807690544490179709803177425355852401229...

%e A(1/5) = 0.16558333624735433324843855679493132539350188690309...

%e A(1/6) = 0.14230098666491512550971306545368484826875874989347...

%e (End)

%p with(numtheory):

%p 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

%o (PARI) {a(n) = local(A=1); A = sum(k=-n, n, (x - x^k)^k +x*O(x^n)); polcoeff(A, n)}

%o for(n=0, 100, print1(a(n), ", "))

%o (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)}

%o for(n=0, 500, print1(a(n), ", "))

%Y Cf. A260116, A260147, A378582, A379195.

%K sign,easy,changed

%O 0,4

%A _Paul D. Hanna_, Sep 03 2017