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

1, 3, 7, 15, 36, 87, 211, 519, 1285, 3198, 7998, 20079, 50571, 127725, 323367, 820407, 2085306, 5309169, 13537045, 34561890, 88347091, 226079208, 579110262, 1484766015, 3809948461, 9783998877, 25143452881, 64658016249, 166375274790

Offset: 1

Author

Paul D. Hanna, Nov 23 2009

Keywords

• nonn

Examples

			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 +...
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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A004148, variants: A166895, A166897, A166899.

Programs

Mathematica

Table[Sum[(Binomial[n - k, k]^2)*(n/(n - k)), {k, 0, n/2}], {n, 1, 100}] (* G. C. Greubel, Jun 15 2016 *)

PARI

{a(n) = sum(k=0,n\2, binomial(n-k,k)^2 * n/(n-k))}
for(n=1,30,print1(a(n),", "))

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)}
for(n=1,30,print1(a(n),", "))

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)))}
for(n=1,30,print1(a(n),", "))

Formula

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

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

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).

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