例如:A(x)=1+3*x+22*x^2/2!+278*x^3/3!+5128*x^4/4!+125592*x^5/5!+3850000*x^6/6!+142013328*x^7/7!+。。。
使A(x)=exp(2*x*A(x,))/(1-x),其中
exp(2*x*A(x))=1+2*x+16*x^2/2!+212*x^3/3!+4016*x^4/4!+99952*x^5/5!+3096448*x^6/6!+115063328*x^7/7!+。。。
相关表格。
这里说明了例如f.A(x)的另一个有趣的性质。
系数表x^k/k!在1/A(x)^n中开始:
n=1:[1,-3,-4,-44,-736,-16832,-491168,…];
n=2:[1,-6,10,-16,-320,-8064,-249344,…];
n=3:[1,-9,42,-78,-48,-1776,-66528,…];
n=4:[1,-12,92,-392,728,-128,-8960,…];
n=5:[1,-15,160,-1120,4600,-8520,-320,…];
n=6:[1,-18,246,-2424,16104,-64752,119952,…];
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由此我们可以说明,对于n>1,1/A(x)^n中的系数x^k,k=0..n的部分和等于零,如下所示:
n=1:-2=1+-3;
n=2:0=1+-6+10/2!;
n=3:0=1+-9+42/2!+-78/3!;
n=4:0=1+-12+92/2!+-392/3! + 728/4!;
n=5:0=1+-15+160/2!+-1120/3! + 4600/4! + -8520/5!;
n=6:0=1+-18+246/2!+-2424/3! + 16104/4! + -64752/5! + 119952/6!;
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