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A270724
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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.
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1
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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MAPLE
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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 ;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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