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A298300 Analog of Motzkin numbers for Coxeter type D.

%I A298300 #29 Nov 24 2024 00:04:15
%S A298300 1,4,11,31,87,246,699,1996,5723,16468,47533,137567,399073,1160082,
%T A298300 3378483,9855207,28790403,84218052,246651729,723165765,2122391109,
%U A298300 6234634266,18330019029,53932825926,158802303429,467898288676,1379485436579,4069450219561
%N A298300 Analog of Motzkin numbers for Coxeter type D.
%F A298300 a(n) = A002426(n-1) + A290380(n) (the latter being extended by A290380(2)=0).
%F A298300 Conjectural algebraic equation: 3*t+2+(3*t^2+5*t-2)*f(t)+(3*t^3-t^2)*f(t)^2 = 0.
%F A298300 From _Peter Luschny_, Jan 23 2018: (Start)
%F A298300 a(n) = hypergeom([(1-n)/2,1-n/2],[1],4)+(n-2)*hypergeom([1-n/2,3/2-n/2],[2],4).
%F A298300 a(n) = G(n-1,1-n,-1/2) + G(n-2,1-n,-1/2)*(n-2)/(n-1) where G(n,a,x) denotes the n-th Gegenbauer polynomial. (End)
%F A298300 D-finite with recurrence +2*n*a(n) +(-7*n+6)*a(n-1) +9*(n-4)*a(n-3)=0. - _R. J. Mathar_, Jul 27 2022
%p A298300 A298300 := proc(n)
%p A298300     hypergeom([(1-n)/2,1-n/2],[1],4)+(n-2)*hypergeom([1-n/2,3/2-n/2],[2],4);
%p A298300     simplify(%) ;
%p A298300 end proc:
%p A298300 seq(A298300(n),n=2..40) ; # _R. J. Mathar_, Jul 27 2022
%t A298300 b[n_] := Hypergeometric2F1[(1 - n)/2, 1 - n/2, 1, 4];
%t A298300 c[n_] := (n-2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
%t A298300 Table[b[n] + c[n], {n, 2, 29}] (* _Peter Luschny_, Jan 23 2018 *)
%o A298300 (Sage)
%o A298300 def a(n):
%o A298300      return (sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
%o A298300          binomial(n - 2, 2 * i - 2)
%o A298300                  for i in range(1, floor(n / 2) + 1)) +
%o A298300              sum(binomial(n - 1, k) * binomial(n - 1 - k, k)
%o A298300                  for k in range(floor((n - 1) / 2) + 1)))
%Y A298300 Cf. A001006 (type A), A002426 (type B), A290380.
%K A298300 nonn
%O A298300 2,2
%A A298300 _F. Chapoton_, Jan 16 2018