OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
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
EXAMPLE
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 +...
MATHEMATICA
Table[Sum[(Binomial[n - k, k]^2)*(n/(n - k)), {k, 0, n/2}], {n, 1, 100}] (* G. C. Greubel, Jun 15 2016 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 23 2009
STATUS
approved