%I #59 Oct 12 2022 08:44:00

%S 1,2,0,4,-3,6,-3,8,-15,28,-24,12,0,14,-48,96,-95,18,55,20,-180,232,

%T -120,24,-35,76,-168,460,-580,30,515,32,-927,804,-288,456,-497,38,

%U -360,1288,-1169,42,847,44,-2958,3700,-528,48,-2599,148,2526,2772,-5537,54,595,5336,-6930,3820,-840,60,-791,62,-960,6448,-12351,12936,-3167,68,-15435,6648,21365,72,-26646,74,-1368,35776,-23730,8394,-16548,80,7101

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

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

%C Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^(n-1))^n / n = -log(1-x).

%C Whenever a(n+2) is a multiple of n > 7, then a(n+2)/n = -(n+4)/4, with very few exceptions (n = 18, 131, 412, ... and n = 10, a(12) = 0). In particular, when n-1 is a prime of the form p = 4k + 3, then a(p+3) = -(k+2)(p+1) (as compared to a(p) = p+1), except for k = 11, 16, 26, 31, 37, 41, .... What exactly are these exceptions? - _M. F. Hasler_, Oct 10 2017

%H Paul D. Hanna, <a href="/A291937/b291937.txt">Table of n, a(n) for n = 0..6600</a>

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

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

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

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

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

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

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

%F (7) A(x) = Limit_{k->oo} Sum_{n=-oo..+oo} x^(n-k) * (1 - x^n + n*x^(n+k))^n.

%F The l.g.f. L(x) = Sum_{n>=1} a(n) * x^n / n satisfies:

%F (8) L(x) = -1 + Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^n)^n / n.

%F a(p) = p+1 for odd primes p.

%e O.g.f.: A(x) = 1 + 2*x + 4*x^3 - 3*x^4 + 6*x^5 - 3*x^6 + 8*x^7 - 15*x^8 + 28*x^9 - 24*x^10 + 12*x^11 + 14*x^13 - 48*x^14 + 96*x^15 - 95*x^16 + 18*x^17 + 55*x^18 + 20*x^19 - 180*x^20 + 232*x^21 - 120*x^22 + 24*x^23 - 35*x^24 + 76*x^25 - 168*x^26 + 460*x^27 - 580*x^28 + 30*x^29 + 515*x^30 +...

%e where A(x) = P(x) + Q(x) with

%e P(x) = x*(1-x) + 2*x^2*(1-x^2)^2 + 3*x^3*(1-x^3)^3 + 4*x^4*(1-x^4)^4 + 5*x^5*(1-x^5)^5 +...+ n * x^n * (1 + x^n)^n + ...

%e Q(x) = 1/(1-x) - 2*x^2/(1-x^2)^2 + 3*x^6/(1-x^3)^3 - 4*x^12/(1-x^4)^4 + 5*x^20/(1-x^5)^5 + ... + -(-1)^n * n * x^(n^2-n) / (1 - x^n)^n + ...

%e Explicitly,

%e P(x) = x + x^2 + 3*x^3 + 5*x^5 - x^6 + 7*x^7 - 8*x^8 + 18*x^9 - 15*x^10 + 11*x^11 - 3*x^12 + 13*x^13 - 35*x^14 + 65*x^15 - 64*x^16 + 17*x^17 + 27*x^18 + 19*x^19 - 126*x^20 + 168*x^21 - 99*x^22 + 23*x^23 - 16*x^24 + 50*x^25 - 143*x^26 + 351*x^27 - 413*x^28 + 29*x^29 + 340*x^30 + ...

%e Q(x) = 1 + x - x^2 + x^3 - 3*x^4 + x^5 - 2*x^6 + x^7 - 7*x^8 + 10*x^9 - 9*x^10 + x^11 + 3*x^12 + x^13 - 13*x^14 + 31*x^15 - 31*x^16 + x^17 + 28*x^18 + x^19 - 54*x^20 + 64*x^21 - 21*x^22 + x^23 - 19*x^24 + 26*x^25 - 25*x^26 + 109*x^27 - 167*x^28 + x^29 + 175*x^30 + ...

%e Also, A(x) = M(x) + N(x) with

%e M(x) = x^2 + 4*x^4*(1-x^2) + 9*x^6*(1-x^3)^2 + 16*x^8*(1-x^4)^3 + 25*x^10*(1-x^5)^4 + ... + n^2 * x^(2*n) * (1 - x^n)^(n-1) + ...

%e N(x) = 1/(1-x)^2 - 4*x^2/(1-x^2)^3 + 9*x^6/(1-x^3)^4 - 16*x^12/(1-x^4)^5 + 25*x^20/(1-x^5)^6 + ... + -(-1)^n * n^2 * x^(n^2-n) / (1 - x^n)^(n+1) + ...

%e Explicitly,

%e M(x) = x^2 + 4*x^4 + 5*x^6 + 16*x^8 - 18*x^9 + 25*x^10 - 3*x^12 + 49*x^14 - 100*x^15 + 112*x^16 - 99*x^18 + 234*x^20 - 294*x^21 + 121*x^22 + 56*x^24 - 100*x^25 + 169*x^26 - 648*x^27 + 931*x^28 - 1010*x^30 + ...

%e N(x) = 1 + 2*x - x^2 + 4*x^3 - 7*x^4 + 6*x^5 - 8*x^6 + 8*x^7 - 31*x^8 + 46*x^9 - 49*x^10 + 12*x^11 + 3*x^12 + 14*x^13 - 97*x^14 + 196*x^15 - 207*x^16 + 18*x^17 + 154*x^18 + 20*x^19 - 414*x^20 + 526*x^21 - 241*x^22 + 24*x^23 - 91*x^24 + 176*x^25 - 337*x^26 + 1108*x^27 - 1511*x^28 + 30*x^29 + 1525*x^30 + ...

%e Terms at powers of 2 begin:

%e a(2^n) = [2, 0, -3, -15, -95, -927, -12351, -457215, -137484287, -71927383551, -12774376215944191, -2073810501234874519551, -78004011261694477161745918353407, ...].

%e Terms at powers of 3 begin:

%e a(3^n) = [2, 4, 28, 460, 10774, 80195104, 2894790054826, ..., A292184(n), ...].

%e Terms at powers of 5 begin:

%e a(5^n) = [2, 6, 76, 379626, 1259880626, 4828768869002981409762696876, ...].

%t terms = 200; Sum[n*x^n*(1 - x^n)^n, {n, -terms, terms}] + O[x]^terms //

%t CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 11 2017 *)

%o (PARI) {a(n)=my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, k*x^k*(l-x^k)^k), n)} \\ Edited by _M. F. Hasler_, Oct 11 2017

%o (PARI) {a(n) = my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, if(k, k^2 * x^(2*k) * (l - x^k)^(k-1))), n)} \\ Edited by _M. F. Hasler_, Oct 11 2017

%o (PARI) {a(n) = my(x='x+O('x^(2*n+2))); polcoeff(sum(k=-n-2, sqrtint(2*n)+2, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^k), n)} \\ Edited by _M. F. Hasler_, Oct 11 2017

%o (PARI) {a(n) = my(x='x+O('x^(2*n+2))); polcoeff( sum(k=-n-2, sqrtint(2*n), if(k, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^(k+1) )), n)} \\ Edited by _M. F. Hasler_, Oct 11 2017

%o for(n=0, 80, print1(a(n), ", "))

%o (PARI) A291937_vec(n)={my(x='x+O('x^(2*n+2))); -Vec(sum(k=-n-2, sqrtint(2*n), if(k,(-1)^k*k*x^(k^2-k)/(1-x^k)^(k+1))))[1..n+1]} \\ In case several values in a(0..n) are required, it is most efficient to compute the whole vector at once. E.g., sum(n=0..150,a(n)) takes ~ 10 sec., vecsum(A291937_vec(150)) takes ~ 0.1 sec. - _M. F. Hasler_, Oct 11 2017

%Y Cf. A292184, A262007, A292177, A293129.

%K sign,look

%O 0,2

%A _Paul D. Hanna_, Sep 06 2017