A270724 - OEIS

%I #18 Oct 07 2016 10:39:53

%S 1,3,5,10,20,42,93,213,504,1221,3014,7553,19158,49087,126845,330174,

%T 864884,2278138,6030218,16031950,42790362,114616360,307996874,

%U 830084080,2243193198,6076953906,16500486744,44897830740,122406923038,334333367602

%N 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.

%F G.f.: ((-x^2+x+1)*(1-sqrt(1-(4*x^2*(x+1))/(1-x))))/(2*x^2*(1-x^2)).

%F 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).

%F 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

%p A270724 := proc(n)

%p     a := 0 ;

%p     for k from 0 to n/2 do

%p         for i from 0 to n-2*k do

%p             a := a+binomial(k+1,n-2*k-i)*binomial(k+i,k)/(k+1)*A000108(k) ;

%p         end do:

%p     end do:

%p     %*(n+2)/2 ;

%p end proc: # _R. J. Mathar_, Oct 07 2016

%t 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 *)

%t 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 *)

%o (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));

%o (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

%Y Cf. A000108, A113413, A270715.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Mar 22 2016