A206152 - OEIS

%I #13 Jan 29 2019 09:10:59

%S 1,2,10,326,64066,111968752,1091576358244,106664423412770932,

%T 67305628532703785062402,329378455047908259704557301276,

%U 15577435010841058543979449475481629020,4149966977623235242137197627437116176363522092

%N a(n) = Sum_{k=0..n} binomial(n,k)^(n+k).

%C Ignoring initial term, equals the logarithmic derivative of A206151.

%H Seiichi Manyama, <a href="/A206152/b206152.txt">Table of n, a(n) for n = 0..47</a>

%F Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274005711865243..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - _Vaclav Kotesovec_, Mar 03 2014

%e L.g.f.: L(x) = 2*x + 10*x^2/2 + 326*x^3/3 + 64066*x^4/4 + 111968752*x^5/5 +...

%e where exponentiation yields A206151:

%e exp(L(x)) = 1 + 2*x + 7*x^2 + 120*x^3 + 16257*x^4 + 22426576*x^5 +...

%e Illustration of initial terms:

%e a(1) = 1^1 + 1^2 = 2;

%e a(2) = 1^2 + 2^3 + 1^4 = 10;

%e a(3) = 1^3 + 3^4 + 3^5 + 1^6 = 326;

%e a(4) = 1^4 + 4^5 + 6^6 + 4^7 + 1^8 = 64066; ...

%t Table[Sum[Binomial[n,k]^(n+k), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(n+k))}

%Y Cf. A206151, A227403, A237421.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012