|
%I #20 Nov 15 2017 21:42:29
%S -2,-4,-6,-12,-14,-22,-30,-40,-38,-44,-64,-54,-40,16,12,54,94,248,370,
%T 672,754,932,1112,1360,2388,2684,3508,4388,4452,4240,7342,7538,9852,
%U 14268,19268,16892,22950,24514,24854,41138,50750,54974,89336,117050,103586,140074,153180,147910,236252,267510,274156,520150,515664,568144,906496,834244,552372,1130104,1142566,933698,1772950,1915700
%N G.f.: Product_{n=-oo..+oo} ( 1 + x^n*(1 - x^n)^n ).
%C Compare g.f. to: Sum_{n=-oo..+oo} x^n*(1 - x^n)^n = 0.
%H Paul D. Hanna, <a href="/A293602/b293602.txt">Table of n, a(n) for n = 1..520</a>
%F G.f.: A(x) = P(x) * Q(x) where
%F P(x) = Product_{n>=0} ( 1 + x^n*(1 - x^n)^n ),
%F Q(x) = Product_{n>=1} ( 1 + (-1)^n * x^(n^2-n)/(1 - x^n)^n ).
%e G.f.: A(x) = -2*x - 4*x^2 - 6*x^3 - 12*x^4 - 14*x^5 - 22*x^6 - 30*x^7 - 40*x^8 - 38*x^9 - 44*x^10 - 64*x^11 - 54*x^12 - 40*x^13 + 16*x^14 + 12*x^15 + 54*x^16 + 94*x^17 + 248*x^18 + 370*x^19 + 672*x^20 + 754*x^21 + 932*x^22 + 1112*x^23 + 1360*x^24 + 2388*x^25 + 2684*x^26 + 3508*x^27 + 4388*x^28 + 4452*x^29 + 4240*x^30 +...
%e where
%e A(x) = P(x) * Q(x) where
%e P(x) = 2 * (1 + x*(1-x)) * (1 + x^2*(1-x^2)^2) * (1 + x^3*(1-x^3)^3) * (1 + x^4*(1-x^4)^4) * (1 + x^5*(1-x^5)^5) *...* ( 1 + x^n*(1 - x^n)^n ) *...
%e Q(x) = (1 - 1/(1-x)) * (1 + x^2/(1-x^2)^2) * (1 - x^6/(1-x^3)^3) * (1 + x^12/(1-x^4)^4) * (1 - x^20/(1-x^5)^5) *...* (1 + (-1)^n * x^(n^2-n)/(1 - x^n)^n ) *...
%e Explicitly,
%e P(x) = 2 + 2*x + 4*x^3 - 2*x^4 + 6*x^6 - 2*x^7 - 10*x^8 + 22*x^10 - 18*x^11 - 36*x^13 - 4*x^14 + 98*x^15 - 10*x^16 - 104*x^17 + 30*x^18 - 108*x^19 - 40*x^20 + 448*x^21 - 16*x^22 - 200*x^23 - 214*x^24 - 148*x^25 - 586*x^26 + 718*x^27 + 1374*x^28 - 224*x^29 + 104*x^30 +...
%e Q(x) = -x - x^2 - 2*x^3 - 2*x^4 - 4*x^5 - 4*x^6 - 6*x^7 - 6*x^8 - 9*x^9 - 6*x^10 - 9*x^11 - 6*x^12 - 4*x^13 + 2*x^14 + 4*x^15 + 23*x^16 + 26*x^17 + 48*x^18 + 72*x^19 + 107*x^20 + 123*x^21 + 195*x^22 + 232*x^23 + 317*x^24 + 376*x^25 + 515*x^26 + 595*x^27 + 817*x^28 + 912*x^29 + 1215*x^30 +...
%t terms = 62; Product[If[n == 0, 2, (1 + x^n*(1 - x^n)^n)], {n, -terms, terms}] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Nov 04 2017 *)
%o (PARI) {a(n) = my(A); A = prod(m=-n-1,n+1, (1 + x^m*(1 - x^m)^m + x*O(x^n)) ); polcoeff(A,n)}
%o for(n=1,60,print1(a(n),", "))
%Y Cf. A293603, A295131.
%K sign,look
%O 1,1
%A _Paul D. Hanna_, Oct 21 2017
|
