O.g.f:A(x)=1+2*x+10*x^2+104*x^3+1772*x^4+42408*x^5+1303504*x^6+48736000*x^7+2139552016*x^8+107629121888*x^9+6094743943584*x^10+。。。
定义说明。
系数表x^k/k!在exp(n*(n+1)*x)/A(x)中开始:
n=0:[1,-2,-12,-432,-32640,-4176000,-804504960,-216834831360,…];
n=1:[1,0,-16,-520,-36432,-4520768,-856647680,-228458074752,…];
n=2:[1,4,0,-648,-46032,-5341824,-974612736,-254049782400,…];
n=3:[1,10,84,0,-56832,-6922368,-1194341760,-299397745152,…];
n=4:[1,18,308,4448,0,-8528000,-1573784960,-376524760,…];
n=5:[1,28,768,20088,444720,0,-1938504960,-502258872960,…];
n=6:[1,40,1584,61560,2286768,72032832,0,-618983309952,…];
n=7:[1,54,2900,154352,8074368,404450176,17201640064,0,…];
n=8:[1,70,4884,339120,23357568,1583068032,102886277760,5682964174848,0,…]。。。
其中主对角线在初始项之后全部为零,说明[x^n]exp(n*(n+1)*x)/A(x)=0,表示n>=0。
逻辑推导。
A’(x)/A(x)=2+16*x+260*x^2+6200*x^3+191832*x^4+7235152*x^5+320372320*x^6+16243028896*x^7+926219213216*x^8+58608051937536*x^9+4072306624576*x^10++A304317型(n) *x^n+。。。