组织结构:A(x)=1+x+2*x^2+11*x^3+130*x^4+2450*x^5+63012*x^6+2040779*x^7+79377914*x^8+359476694*x^9+185457776252*x^10+。。。
定义说明。
系数表x^k/k!在exp(n*(n+1)*x)/A(x)^(n+1
n=0:[1,-1,-2,-48,-2618,-262080,-41718240,-9630270720,…];
n=1:[1,0,-6,-112,-5592,-547968,-86345120,-1980990912,…];
n=2:[1,3,0,-222,-10728,-958824,-144971712,-32519314080,…];
n=3:[1,8,52,0,-18648,-1693248,-236690784,-50727983616,…];
n=4:[1,15,210,2420,0,-2739720,-399251600,-80125144800,…];
n=5:[1,24,558,12192,221184,0,-616918320,-131299591680,…];
n=6:[1,35,1204,40278,1272768,33597312,0,-196436730672,…];
n=7:[1,48,2280,106688,4869552,210771456,7654459648,0,…]。。。
其中,主对角线在初始项之后全部为零,说明了当n>0时,[x^n]exp(n*(n+1)*x)/A(x)^(n+1)=0。
相关系列。
定义B(x)=A(x/B(x)),从
B(x)=1+x+x^2+7*x^3+93*x^4+1859*x^5+49357*x^6+1629227*x^7+64149805*x^8+2929386667*x^9++A337457型(n) *x^n+。。。
然后是系数表x^k/k!在exp(n*(n-1)*x/B(x))中开始:
n=0:[1,0,0,0,0,0,1,0,0,0…];
n=1:[1,0,0,0,0,0:0,…];
n=2:[1,2,0,-16,-320,-21888,-2648576,-494325760,…];
n=3:[1,6,24,0,-1728,-88704,-9621504,-1715198976,…];
n=4:[1,121220664,0,-281088,-26873856,-4328017920,…];
n=5:[1,20,360,5600,65920,0,-66944000,-10207436800,…];
n=6:[1,30,840,21600,492480,8784000,0,-22098355200,…];
n=7:[1,42,1680,63504,2237760,71229312,1814690304,0,…]。。。
其中,主对角线在初始项之后全部为零,说明当n>0时,[x^n]exp(n*(n-1)*x/B(x))=0。
还要注意,B(x)=x/系列_反转(x*A(x))和A(x)=B(x*A(x)。
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