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A290380
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Analog of Motzkin sums for Coxeter type D.
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3
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1, 4, 12, 36, 105, 306, 889, 2584, 7515, 21880, 63778, 186132, 543855, 1590876, 4658580, 13655472, 40065243, 117654876, 345786396, 1017040380, 2993498739, 8816790906, 25984489545, 76625467128, 226085062525, 667415280376, 1971209865654, 5824651789852
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OFFSET
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3,2
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COMMENTS
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See proposition 3.3 of the Athanasiadis-Savvidou reference.
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n/2} (n-2)/i*binomial(2*i-2, i-1)*binomial(n-2, 2*i-2).
a(n) = (n - 2)*hypergeom([1 - n/2, 3/2 - n/2], [2], 4).
a(n) = (-1)^n (n - 2)*hypergeom([2 - n, 3/2], [3], 4).
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MAPLE
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a:= proc(n) option remember; `if`(n<5, [0$2, 1, 4][n],
((n-2)*(2*n-3)*(n-4)*a(n-1)+3*(n-2)*(n-3)^2*
a(n-2))/((n-3)*(n-4)*n))
end:
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MATHEMATICA
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Table[Sum[(n - 2)/i*Binomial[2i - 2, i - 1] Binomial[n - 2, 2i - 2], {i, n/2}], {n, 3, 50}] (* Indranil Ghosh, Jul 29 2017 *)
a[n_] := (n - 2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
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PROG
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(Sage)
return sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
binomial(n - 2, 2 * i - 2)
for i in range(1, n // 2 + 1))
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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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