OFFSET
0,2
FORMULA
G.f.: ((-x^2+x+1)*(1-sqrt(1-(4*x^2*(x+1))/(1-x))))/(2*x^2*(1-x^2)).
a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))*binomial(2*k,k)/(k+1)^2).
Conjecture: (n+2)*a(n) +(-n-2)*a(n-1) +(-7*n+6)*a(n-2) +10*a(n-3) +(13*n-32)*a(n-4) +(5*n-32)*a(n-5) +(-11*n+52)*a(n-6) +4*(-n+6)*a(n-7) +4*(n-7)*a(n-8)=0. - R. J. Mathar, Oct 07 2016
MAPLE
A270724 := proc(n)
a := 0 ;
for k from 0 to n/2 do
for i from 0 to n-2*k do
a := a+binomial(k+1, n-2*k-i)*binomial(k+i, k)/(k+1)*A000108(k) ;
end do:
end do:
%*(n+2)/2 ;
end proc: # R. J. Mathar, Oct 07 2016
MATHEMATICA
Table[((n + 2)/2) Sum[Sum[(Binomial[k + 1, n - 2 k - i] Binomial[k + i, k]) Binomial[2 k, k]/(k + 1)^2, {i, 0, n - 2 k}], {k, 0, n/2}], {n, 0, 29}] (* or *)
CoefficientList[Series[((-x^2 + x + 1) (1 - Sqrt[1 - (4 x^2 (x + 1))/(1 - x)]))/(2 x^2*(1 - x^2)), {x, 0, 29}], x] (* Michael De Vlieger, Mar 25 2016 *)
PROG
(Maxima) a(n):=((n+2)/2)*(sum(sum(binomial(k+1, n-2*k-i)*binomial(k+i, k), i, 0, n-2*k)/(k+1)^2*binomial(2*k, k), k, 0, n/2));
(PARI) x='x+O('x^200); Vec(((-x^2+x+1)*(1-sqrt(1-(4*x^2*(x+1))/(1-x))))/(2*x^2*(1-x^2))) \\ Altug Alkan, Mar 22 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 22 2016
STATUS
approved