A245110 G.f.: Sum_{n>=0} ( exp(-1/(1-n*x)) / (1-n*x)^n ) / n!.
1, 1, 4, 23, 161, 1302, 11810, 117889, 1277890, 14894043, 185226966, 2442933979, 33998594943, 497207012018, 7613797641286, 121711037138949, 2025687745708717, 35020194893837462, 627586143525936866, 11636932722633705392, 222893347544826491780, 4403534468187986689781
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 161*x^4 + 1302*x^5 + 11810*x^6 +... where A(x) = exp(-1) + exp(-1/(1-x))/(1-x) + (exp(-1/(1-2*x))/(1-2*x)^2)/2! + (exp(-1/(1-3*x))/(1-3*x)^3)/3! + (exp(-1/(1-4*x))/(1-4*x)^4)/4! + (exp(-1/(1-5*x))/(1-5*x)^5)/5! + (exp(-1/(1-6*x))/(1-6*x)^6)/6! + (exp(-1/(1-7*x))/(1-7*x)^7)/7! + (exp(-1/(1-8*x))/(1-8*x)^8)/8! +... simplifies to a power series in x with integer coefficients.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..150
Programs
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PARI
/* From definition (requires setting suitable precision) */ \p100 {a(n)=local(A=1+x, X=x+x*O(x^n)); A=suminf(k=0, exp(-1/(1-k*X))/(1-k*X)^k/k!); round(polcoeff(A, n))} for(n=0, 30, print1(a(n), ", ")) -
PARI
/* From a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) */ {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!} {a(n)=if(n==0,1,sum(k=1,n,Stirling2(n, k) * binomial(n+k-1, k-1)))} for(n=0, 30, print1(a(n),", ")) -
PARI
/* As row sums of triangle A245111: */ {A245111(n,k)=local(A=1+x*y); A=sum(k=0, n, 1/(1-k*x+x*O(x^n))^k*y^k/k!*exp(-y/(1-k*x+x*O(x^n))+y*O(y^n))); polcoeff(polcoeff(A, n,x),k,y)} {a(n) = sum(k=0,n, A245111(n,k))} /* Print Initial Rows of Triangle A245111: */ {for(n=0, 10, for(k=0,n, print1(A245111(n,k),", "));print(""))} /* Row Sums yield A245110: */ for(n=0, 30, print1(a(n),", "))
Formula
a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) for n>0 with a(0)=1.
Row sums of Triangle A245111.
Comments