A166899 - OEIS

%I #14 Nov 20 2024 04:58:24

%S 1,3,25,111,456,2697,15961,86247,495781,3003738,17946798,107667969,

%T 660458787,4081397547,25274724105,157744019799,991384251102,

%U 6254115981009,39613066988527,252017709962526,1608980424431755

%N a(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k), n>=1.

%H G. C. Greubel, <a href="/A166899/b166899.txt">Table of n, a(n) for n = 1..501</a> [Offset shifted by _Georg Fischer_, Nov 20 2024]

%F Logarithmic derivative of A166898.

%F a(n) ~ 5^(3/4) * phi^(4*n+3) / (2^(5/2) * Pi^(3/2) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 27 2017

%e L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 +...

%e exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...+ A166898(n)*x^n +...

%t Table[Sum[Binomial[n - k, k]^4 *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 50}] (* _G. C. Greubel_, May 27 2016 *)

%o (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^4*n/(n-k))

%Y Cf. A166898, variants: A167539, A166895, A166897.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 23 2009

%E Offset changed to 1 by _Georg Fischer_, Nov 20 2024