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A203309 Vandermonde determinant of the first n triangular numbers.

%I A203309 #30 Aug 30 2023 02:02:02
%S A203309 1,1,2,30,7560,57153600,20369543040000,495474875767872000000,
%T A203309 1124860755259775229696000000000,
%U A203309 312577210159744965479388971827200000000000,13502658421660070413446616883411391637094400000000000000
%N A203309 Vandermonde determinant of the first n triangular numbers.
%C A203309 Each term divides its successor, as in A203310.
%H A203309 G. C. Greubel, <a href="/A203309/b203309.txt">Table of n, a(n) for n = 0..31</a>
%F A203309 a(n) ~ 2^(n*(n + 5)/2 - 7/24) * Pi^((n-1)/2) * n^(n^2 - n/2 - 37/24) / (sqrt(A) * exp(n*(3*n - 1)/2 - 1/24)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jan 25 2019
%F A203309 From _G. C. Greubel_, Aug 29 2023: (Start)
%F A203309 a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+1)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).
%F A203309 a(n) = (1/2)^binomial(n,2)*(BarnesG(n+1))^2*Product_{k=2..n} binomial(2*k, k+1).
%F A203309 a(n) = Product_{k=1..n-1} k!*(2*k+2)!/(2^k*(k+2)!). (End)
%p A203309 with(LinearAlgebra):
%p A203309 a:= n-> Determinant(VandermondeMatrix([i*(i+1)/2$i=1..n])):
%p A203309 seq(a(n), n=0..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203309 (* First program *)
%t A203309 f[j_]:= j*(j+1)/2; z = 15;
%t A203309 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203309 Table[v[n], {n,0,z}]           (* A203309 *)
%t A203309 Table[v[n+1]/v[n], {n,0,z}]    (* A203310 *)
%t A203309 (* Second program *)
%t A203309 Table[(2^(n+3)/Pi)^(n/2)*BarnesG[n+1]*BarnesG[n+3/2]/(Gamma[n+ 2]*BarnesG[3/2]), {n,0,20}] (* _G. C. Greubel_, Aug 29 2023 *)
%o A203309 (Python)
%o A203309 from operator import mul
%o A203309 from functools import reduce
%o A203309 def f(n): return n*(n + 1)//2
%o A203309 def v(n): return 1 if n==1 else reduce(mul, [f(k) - f(j) for k in range(2, n + 1) for j in range(1, k)])
%o A203309 print([v(n) for n in range(1, 11)]) # _Indranil Ghosh_, Jul 24 2017
%o A203309 (Magma) F:= Factorial; [1] cat [(&*[(F(k)*F(2*k+2))/(2^k*F(k+2)): k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 29 2023
%o A203309 (SageMath) f=factorial; [product((f(j)*f(2*j+2))/(2^j*f(j+2)) for j in range(n)) for n in range(21)] # _G. C. Greubel_, Aug 29 2023
%Y A203309 Cf. A074962, A203310, A203467.
%K A203309 nonn
%O A203309 0,3
%A A203309 _Clark Kimberling_, Jan 01 2012
%E A203309 a(0)=1 prepended by _Alois P. Heinz_, Aug 29 2023