O、 g.f.:A(x)=1+2*x+4*x^3-3*x^4+6*x^5-3*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 x^14+96*x^15-96*x^15-95*x^16+18*x^17+55*x^18+55*x 18+20*x^19-180*x^20+232*x^21-120*x x^22+24*24*x^23-35*x x^24+24*x^23-35*x x^24+24*x^23-35*x^24+24*x^24*x^24*x^26+460*x^27-580*x^28+30*x^29+515*x^30+。。。
式中A(x)=P(x)+Q(x),其中
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+。。。
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+。。。
明确地,
P(x)P(x)P=x+x^2+3*x^3+5*x^5-x x^6+7*x^7-8*x^8+18*x^9-15*x x^10+11*x^11-3*x^12+13*x^13-35*x^14+65*x^15-64*x x^16+16+17*x^17+27*x^18+18+19*x^19-126*x^20+168*x^21-99*x^22+23*x^23-23*x^23-16*x^24+50*x^25-25-143*x x^26+351*x x^16*x^24+50*x 25-143*x*x x x x^26+351*^27-413*x^28+29*x^29+340*x^30+。。。
Q(x)Q(x)=1+x-x x^2+x^3-3*x^4+x^5-2*x^6+x^7-7*7*x^8+10*x^9-9*x x^9-9*x^10+x^11+11+3*x^12+x^13-13*x^14+31*x^15-31*x ^15*31*x^16*x^17+28*x^18+18+x^19-54*x x^20+54*x 20+x^21-21*x^22+x^22+x^23-19*x^24+26*x^25-25*x 25*x^26+25*x^26*x^25*x^26+109*x^27*x^27*x 28+x^29+175*x^30+。。。
另外,A(x)=M(x)+N(x),其中
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)+。。。
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)+。。。
明确地,
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+。。。
N(x)=1+2*x 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+x 12+14*x^13-97*x^14*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^23-24*x^24*x^14*x^14*x^14*x^20+526*x^21-241*x^22+24*x^22+24*x^23+176*x^25-337*x^26+1108*x^27-1511*x^28+30*x^29+1525*x^30+。。。
2次方的任期开始:
a(2^n)=[2,0,-3,-15,-95,-927,-12351,-457215,-137484287,-71927383551,-12774376215944191,-207385012348744519551,-7800411261694477161745918353407,…]。
3次方的任期开始:
a(3^n)=[2,4,28,460,10774,80195104,2894790054826。。。,A292184(n) ,…]。
5次方的任期开始:
a(5^n)=[2,6,76,379626,1259880626,4828768869002981409762696876,…]。
|