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
Keywords
Programs
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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
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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 *) -
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));
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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
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