cp's OEIS Frontend

A167539 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2 * n/(n-k), n>=1.

%I A167539 #24 Nov 27 2017 13:29:38
%S A167539 1,3,7,15,36,87,211,519,1285,3198,7998,20079,50571,127725,323367,
%T A167539 820407,2085306,5309169,13537045,34561890,88347091,226079208,
%U A167539 579110262,1484766015,3809948461,9783998877,25143452881,64658016249,166375274790
%N A167539 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2 * n/(n-k), n>=1.
%H A167539 G. C. Greubel, <a href="/A167539/b167539.txt">Table of n, a(n) for n = 1..1000</a>
%F A167539 L.g.f.: Log((1 - x - x^2 - sqrt((1+x+x^2)*(1-3*x+x^2)))/(2*x^3)) = Sum_{n>=1} a(n)*x^n/n. - _Paul D. Hanna_, Jul 19 2015
%F A167539 L.g.f.: -Log((1 - x - x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)))/2) = Sum_{n>=1} a(n)*x^n/n. (Minor simplification of the l.g.f. given above.) - _Petros Hadjicostas_, Oct 25 2017
%F A167539 a(n) = Sum_{k=0..n-1} Sum_{j=0..k} C(n-k+j,n-k)*n/(n-k+j) * C(n-k,k-j)*C(k-j,j).
%F A167539 a(n) ~ 5^(1/4) * phi^(2*n + 1) / (2*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 27 2017
%e A167539 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 36*x^5/5 + 87*x^6/6 +...
%e A167539 exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A004148(n+1)*x^n/n +...
%t A167539 Table[Sum[(Binomial[n - k, k]^2)*(n/(n - k)), {k, 0, n/2}], {n, 1, 100}] (* _G. C. Greubel_, Jun 15 2016 *)
%o A167539 (PARI) {a(n) = sum(k=0,n\2, binomial(n-k,k)^2 * n/(n-k))}
%o A167539 for(n=1,30,print1(a(n),", "))
%o A167539 (PARI) {a(n) = n * polcoeff( log( (1 - x - x^2 - sqrt((1+x+x^2)*(1-3*x+x^2) +x^6*O(x^n) )) / (2*x^3) ), n)}
%o A167539 for(n=1,30,print1(a(n),", "))
%o A167539 (PARI) {a(n) = sum(k=0,n-1,sum(j=0,k, binomial(n-k+j,n-k)*n/(n-k+j) * binomial(n-k,k-j)*binomial(k-j,j)))}
%o A167539 for(n=1,30,print1(a(n),", "))
%Y A167539 Cf. A004148, variants: A166895, A166897, A166899.
%K A167539 nonn
%O A167539 1,2
%A A167539 _Paul D. Hanna_, Nov 23 2009