A218667
O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).
Original entry on oeis.org
1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301
Offset: 0
O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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/* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-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, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))
A245110
G.f.: Sum_{n>=0} ( exp(-1/(1-n*x)) / (1-n*x)^n ) / n!.
Original entry on oeis.org
1, 1, 4, 23, 161, 1302, 11810, 117889, 1277890, 14894043, 185226966, 2442933979, 33998594943, 497207012018, 7613797641286, 121711037138949, 2025687745708717, 35020194893837462, 627586143525936866, 11636932722633705392, 222893347544826491780, 4403534468187986689781
Offset: 0
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.
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/* 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), ", "))
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/* 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),", "))
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/* 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),", "))
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