The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A245110 G.f.: Sum_{n>=0} ( exp(-1/(1-n*x)) / (1-n*x)^n ) / n!. 4

%I

%S 1,1,4,23,161,1302,11810,117889,1277890,14894043,185226966,2442933979,

%T 33998594943,497207012018,7613797641286,121711037138949,

%U 2025687745708717,35020194893837462,627586143525936866,11636932722633705392,222893347544826491780,4403534468187986689781

%N G.f.: Sum_{n>=0} ( exp(-1/(1-n*x)) / (1-n*x)^n ) / n!.

%C Compare the g.f. to: Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).

%H Paul D. Hanna, <a href="/A245110/b245110.txt">Table of n, a(n) for n = 0..150</a>

%F a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) for n>0 with a(0)=1.

%F Row sums of Triangle A245111.

%e G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 161*x^4 + 1302*x^5 + 11810*x^6 +...

%e where

%e A(x) = exp(-1) + exp(-1/(1-x))/(1-x) + (exp(-1/(1-2*x))/(1-2*x)^2)/2!

%e + (exp(-1/(1-3*x))/(1-3*x)^3)/3! + (exp(-1/(1-4*x))/(1-4*x)^4)/4!

%e + (exp(-1/(1-5*x))/(1-5*x)^5)/5! + (exp(-1/(1-6*x))/(1-6*x)^6)/6!

%e + (exp(-1/(1-7*x))/(1-7*x)^7)/7! + (exp(-1/(1-8*x))/(1-8*x)^8)/8! +...

%e simplifies to a power series in x with integer coefficients.

%o (PARI) /* From definition (requires setting suitable precision) */ \p100

%o {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))}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* From a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) */

%o {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}

%o {a(n)=if(n==0,1,sum(k=1,n,Stirling2(n, k) * binomial(n+k-1, k-1)))}

%o for(n=0, 30, print1(a(n),", "))

%o (PARI) /* As row sums of triangle A245111: */

%o {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)}

%o {a(n) = sum(k=0,n, A245111(n,k))}

%o /* Print Initial Rows of Triangle A245111: */

%o {for(n=0, 10, for(k=0,n, print1(A245111(n,k),", "));print(""))}

%o /* Row Sums yield A245110: */

%o for(n=0, 30, print1(a(n),", "))

%Y Cf. A245109, A245111.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 12 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 27 06:18 EST 2022. Contains 350601 sequences. (Running on oeis4.)