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A290380 Analog of Motzkin sums for Coxeter type D.

%I A290380 #31 Jun 21 2024 14:18:04
%S A290380 1,4,12,36,105,306,889,2584,7515,21880,63778,186132,543855,1590876,
%T A290380 4658580,13655472,40065243,117654876,345786396,1017040380,2993498739,
%U A290380 8816790906,25984489545,76625467128,226085062525,667415280376,1971209865654,5824651789852
%N A290380 Analog of Motzkin sums for Coxeter type D.
%C A290380 See proposition 3.3 of the Athanasiadis-Savvidou reference.
%H A290380 Christos A. Athanasiadis and Christina Savvidou, <a href="https://www.emis.de/journals/SLC/wpapers/s66athasav.html">The Local h-Vector of the Cluster Subdivision of a Simplex</a>, Séminaire Lotharingien de Combinatoire 66 (2012), Article B66c.
%F A290380 a(n) = Sum_{i=1..n/2} (n-2)/i*binomial(2*i-2, i-1)*binomial(n-2, 2*i-2).
%F A290380 From _Peter Luschny_, Jan 23 2018: (Start)
%F A290380 a(n) = (n - 2)*hypergeom([1 - n/2, 3/2 - n/2], [2], 4).
%F A290380 a(n) = (-1)^n (n - 2)*hypergeom([2 - n, 3/2], [3], 4).
%F A290380 a(n) = (n - 2)*A001006(n-2). (End)
%F A290380 G.f.: ((x-2)*sqrt(-3*x^2-2*x+1)-3*x^2-3*x+2)/(2*sqrt(-3*x^2-2*x+1)). - _Vladimir Kruchinin_, Jun 21 2024
%p A290380 a:= proc(n) option remember; `if`(n<5, [0$2, 1, 4][n],
%p A290380       ((n-2)*(2*n-3)*(n-4)*a(n-1)+3*(n-2)*(n-3)^2*
%p A290380       a(n-2))/((n-3)*(n-4)*n))
%p A290380     end:
%p A290380 seq(a(n), n=3..35);  # _Alois P. Heinz_, Jul 28 2017
%t A290380 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 *)
%t A290380 a[n_] := (n - 2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
%t A290380 Table[a[n], {n, 3, 30}] (* _Peter Luschny_, Jan 23 2018 *)
%o A290380 (Sage)
%o A290380 def A290380(n):
%o A290380     return sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
%o A290380                binomial(n - 2, 2 * i - 2)
%o A290380                for i in range(1, n // 2 + 1))
%Y A290380 Cf. A001006, A005043 (type A), A246437 (type B).
%K A290380 nonn
%O A290380 3,2
%A A290380 _F. Chapoton_, Jul 28 2017