A217901 - OEIS

%I #27 Nov 10 2017 18:13:49

%S 1,2,10,106,1736,38414,1073178,36281032,1441336688,65849949118,

%T 3403003693310,196336487214234,12513043615743360,873250527590532680,

%U 66241197525447027832,5427563864923583687376,477771405475710621697632,44970647131664087237328798,4507506792104658670610331462

%N O.g.f.: Sum_{n>=0} 2*n^n * (n+2)^(n-1) * exp(-n*(n+2)*x) * x^n / n!.

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

%C 1 = Sum_{n>=0} 2*(n+2)^(n-1) * exp(-(n+2)*x) * x^n/n!.

%H G. C. Greubel, <a href="/A217901/b217901.txt">Table of n, a(n) for n = 0..345</a>

%F a(n) = 1/n! * Sum_{k=0..n} 2*(-1)^(n-k)*binomial(n,k) * k^n * (k+2)^(n-1).

%F a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(k+2)^(k-1)*x^k / (1 + k*(k+2)*x)^(k+1).

%F a(n) = [x^n] 1 + 2*x*(1+2*x)^(n-1) / Product_{k=1..n} (1-k*x).

%F a(n) = [x^n] 1 + 2*x*(1-2*x)^(n-1) / Product_{k=1..n} (1-(k+2)*x).

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

%e O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 106*x^3 + 1736*x^4 + 38414*x^5 + 1073178*x^6 +...

%e where

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

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

%t Flatten[{1,Table[Sum[Binomial[n-1,j]*2^(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,2*m^m*(m+2)^(m-1)*x^m*exp(-m*(m+2)*x+x*O(x^n))/m!),n)}

%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, 2*k^k*(k+2)^(k-1)*x^k/(1+k*(k+2)*x +x*O(x^n))^(k+1)), n)}

%o (PARI) {a(n)=1/n!*sum(k=0,n, 2*(-1)^(n-k)*binomial(n,k)*k^n*(k+2)^(n-1))}

%o (PARI) {a(n)=polcoeff(1+2*x*(1+2*x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(1+2*x*(1-2*x)^n/prod(k=0, n, 1-(k+2)*x +x*O(x^n)), n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A217899, A217900, A217902, A217903, A217904, A217905.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 14 2012