例如:A(x)=1+x+7*x^2/2!+95*x^3/3!+1969*x^4/4!+55201*x ^ 5/5!+。。。
哪里
1/(cos(x)-sin(x))=1+x+3*x^2!+11*x^3/3!+57*x^4/4!+361*x^5/5!+2763*x^6/6!+24611*x^7/7!++A001586号(n) *x^n/n!+。。。
x^n/n!的系数!以G(x)=1/(cos(x)-sin(x))的奇次幂开始:
G^1:[(1),1,3,11,57,361,2763,24611。。。,A001586号(n) ,…];
G^3:[1,(3),15,93,705,6243,63375,724413,…];
G^5:[1,5,(35),295,2905,32525,407435,5638495,…];
G^7:[1,7,63,(665),8001,107527,1592703,25738265,…];
G^9:[1,9,99,1251,(17721),276849,4716459,86873211,…];
G^11:[1,11,143,2101,34177,(607211),11668943,240764821,…];
G^13:[1,13,195,3263,59865,1190293,(25432875),580193783,…];
G^15:[1,15,255,4785,97665,2146575,50429055,(1259025585),…]。。。
其中括号中的系数构成该序列的初始项:
[1/1, 3/3, 35/5, 665/7, 17721/9, 607211/11, 25432875/13, 1259025585/15, ...].