A344500 a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.
1, 1, 2, 7, 21, 66, 216, 715, 2395, 8101, 27598, 94568, 325612, 1125632, 3904512, 13583195, 47373255, 165585883, 579907758, 2034443127, 7148313381, 25151582046, 88607951512, 312518438532, 1103393962996, 3899415207676, 13792683831176, 48825746365672, 172971084083752
Offset: 0
Keywords
Programs
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Maple
a := n -> add((if n + j mod 2 = 1 then 0 else binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1) fi), j = 0..n): seq(a(n), n = 0..28);
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Mathematica
Table[Sum[(1 + (-1)^(n+j))/2 * Binomial[n, j] * Binomial[n+1, (n-j)/2] * (j+1)/(n+1), {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 21 2024 *)
Formula
a(n) = Sum_{j=0..n} even(n + j)*binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1), where even(k) = 1 if k is even and otherwise 0.
From Vaclav Kotesovec, Apr 21 2024: (Start)
Recurrence: 2*(n+1)*(2*n + 1)*(13*n^3 - 17*n^2 - 12*n + 10)*a(n) = (143*n^5 - 44*n^4 - 525*n^3 + 320*n^2 + 94*n - 60)*a(n-1) + 4*(n-1)*(26*n^4 + 5*n^3 - 108*n^2 + 7*n + 22)*a(n-2) + 16*(n-2)*(n-1)*(13*n^3 + 22*n^2 - 7*n - 6)*a(n-3).
a(n) ~ sqrt((247 - 9131*13^(2/3)/(1788163 + 409728*sqrt(78))^(1/3) + (13*(1788163 + 409728*sqrt(78)))^(1/3))/13) * ((11 + (1/3)*(172017 - 16848*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^n / (sqrt(Pi*n) * 2^(2*n + 3) * 3^(n + 1/2))). (End)