|
%I #10 Jul 30 2014 16:43:19
%S 1,0,1,1,4,13,46,181,778,3585,17566,91171,499324,2873839,17317743,
%T 108933098,713481122,4855161425,34257461754,250177938679,
%U 1887886966690,14699340919293,117933068390123,973776266303732,8265721830953558,72052688932613079,644393453082317301
%N O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).
%C Compare g.f. to the curious identity:
%C 1/(1+x^2) = Sum_{n>=0} (1-n*x)^n * x^n/n! * exp(-x*(1-n*x)).
%H Vaclav Kotesovec, <a href="/A218667/b218667.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(n-1, k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jul 30 2014
%F Antidiagonal sums of Triangle A245111.
%e O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
%e where
%e 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)) +...
%e simplifies to a power series in x with integer coefficients.
%o (PARI) {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)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-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, k) * binomial(n-1, k-1)))}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A245111, A245110, A218668, A218669, A218670, A185040, A217900.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Nov 04 2012
|
