A270724 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))/(k+1)*C(k)), where C(k) is Catalan numbers.

1, 3, 5, 10, 20, 42, 93, 213, 504, 1221, 3014, 7553, 19158, 49087, 126845, 330174, 864884, 2278138, 6030218, 16031950, 42790362, 114616360, 307996874, 830084080, 2243193198, 6076953906, 16500486744, 44897830740, 122406923038, 334333367602

Offset

0,2

Links

Table of n, a(n) for n=0..29.

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

Cf. A000108, A113413, A270715.

Sequence in context: A068013 A342762 A270737 * A018104 A134365 A076862

Adjacent sequences: A270721 A270722 A270723 * A270725 A270726 A270727

Keyword

nonn

Author

Vladimir Kruchinin, Mar 22 2016

Status

approved