A217904 - OEIS

%I #25 Jan 26 2017 02:41:15

%S 1,5,40,550,11000,285380,9064560,340521520,14773539200,727281054640,

%T 40072285049600,2444188361990880,163550098793059200,

%U 11915396563502988800,939110495156447488000,79629365649015094272000,7229173136192077603737600,699726658343948617515436800

%N O.g.f.: Sum_{n>=0} 5*n^n*(n+5)^(n-1) * exp(-n*(n+5)*x) * x^n / n!.

%C Compare the g.f. to the LambertW identity:

%C 1 = Sum_{n>=0} 5*(n+5)^(n-1) * exp(-(n+5)*x) * x^n/n!.

%C From _Vaclav Kotesovec_, May 22 2014: (Start)

%C Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} p*(-1)^(n-k) * binomial(n,k) * k^n * (k+p)^(n-1) = Sum_{j=0..n-1} binomial(n-1,j) * p^(n-j) * StirlingS2(n+j,n).

%C a(n) ~ p * 2^(2*n-3/2+p/2) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...

%C (End)

%H G. C. Greubel, <a href="/A217904/b217904.txt">Table of n, a(n) for n = 0..340</a>

%F a(n) = 1/n! * Sum_{k=0..n} 5*(-1)^(n-k)*binomial(n,k) * k^n * (k+5)^(n-1).

%F a(n) = 1/n! * [x^n] Sum_{k>=0} 5*k^k*(k+5)^(k-1)*x^k / (1 + k*(k+5)*x)^(k+1).

%F a(n) = [x^n] 1 + 5*x*(1+5*x)^(n-1) / Product_{k=1..n} (1-k*x).

%F a(n) = [x^n] 1 + 5*x*(1-5*x)^(n-1) / Product_{k=1..n} (1-(k+5)*x).

%F a(n) ~ 5 * 2^(2*n+1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+5/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014

%e O.g.f.: A(x) = 1 + 5*x + 40*x^2 + 550*x^3 + 11000*x^4 + 285380*x^5 + 9064560*x^6 +...

%e where

%e A(x) = 1 + 5*1^1*6^0*x*exp(-1*6*x) + 5*2^2*7^1*exp(-2*7*x)*x^2/2! + 5*3^3*8^2*exp(-3*8*x)*x^3/3! + 5*4^4*9^3*exp(-4*9*x)*x^4/4! + 5*5^5*10^4*exp(-5*10*x)*x^5/5! +...

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

%t Flatten[{1,Table[Sum[Binomial[n-1,j]*5^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)

%o (PARI) {a(n)=polcoeff(sum(m=0,n,5*m^m*(m+5)^(m-1)*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!),n)}

%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, 5*k^k*(k+5)^(k-1)*x^k/(1+k*(k+5)*x +x*O(x^n))^(k+1)), n)}

%o (PARI) {a(n)=1/n!*sum(k=0,n, 5*(-1)^(n-k)*binomial(n,k)*k^n*(k+5)^(n-1))}

%o (PARI) {a(n)=polcoeff(1+5*x*(1+5*x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(1+5*x*(1-5*x)^n/prod(k=0, n, 1-(k+5)*x +x*O(x^n)), n)}

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

%Y Cf. A217899, A217900, A217901, A217902, A217903, A217905.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2012