|
%I #7 Sep 22 2012 12:19:41
%S 1,3,10,1475,42020826,288102296421912,1549651963209151973674266,
%T 12376315346794076107386866097703962244275,
%U 18103334357369719745485305195095336496837630847237574224638034
%N a(n) = Sum_{k=0..n-1} n*C(n-1,k)^(k^2+k)/(n-k).
%F Equals the logarithmic derivative of A183129.
%F a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)^(k^2+k-1).
%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 1475*x^4/4 + 42020826*x^5/5 +...
%e The l.g.f. equals the series:
%e L(x) = (1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 +...)*x
%e + (1 + 2^2*x + 3^6*x^2 + 4^12*x^3 + 5^20*x^4 + 6^30*x^5 +...)*x^2/2
%e + (1 + 3^2*x + 6^6*x^2 + 10^12*x^3 + 15^20*x^4 + 21^30*x^5 +...)*x^3/3
%e + (1 + 4^2*x + 10^6*x^2 + 20^12*x^3 + 35^20*x^4 + 56^30*x^5 +...)*x^4/4
%e + (1 + 5^2*x + 15^6*x^2 + 35^12*x^3 + 70^20*x^4 + 126^30*x^5 +...)*x^5/5
%e + (1 + 6^2*x + 21^6*x^2 + 56^12*x^3 + 126^20*x^4 + 252^30*x^5 +...)*x^6/6 +...
%e Exponentiation yields the g.f. of A183129:
%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 374*x^4 + 8404542*x^5 + 48017057808567*x^6 + 221378851935038776738734*x^7 +...+ A183129(n)*x^n +...
%t Table[Sum[(n*Binomial[n-1,k]^(k^2+k))/(n-k),{k,0,n-1}],{n,10}] (* _Harvey P. Dale_, Sep 22 2012 *)
%o (PARI) {a(n)=sum(k=0, n-1, n*binomial(n-1, k)^(k^2+k)/(n-k))}
%Y Cf. A183129.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Dec 26 2010
| 