A203309 Vandermonde determinant of the first n triangular numbers.

1, 1, 2, 30, 7560, 57153600, 20369543040000, 495474875767872000000, 1124860755259775229696000000000, 312577210159744965479388971827200000000000, 13502658421660070413446616883411391637094400000000000000

Offset

0,3

Comments

Each term divides its successor, as in A203310.

Links

G. C. Greubel, Table of n, a(n) for n = 0..31

Formula

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

From G. C. Greubel, Aug 29 2023: (Start)

a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+1)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).

a(n) = (1/2)^binomial(n,2)*(BarnesG(n+1))^2*Product_{k=2..n} binomial(2*k, k+1).

a(n) = Product_{k=1..n-1} k!*(2*k+2)!/(2^k*(k+2)!). (End)

Maple

with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([i*(i+1)/2$i=1..n])):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

Mathematica

(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j, k-1}], {k, 2, n}]
Table[v[n], {n, 0, z}]           (* A203309 *)
Table[v[n+1]/v[n], {n, 0, z}]    (* A203310 *)
(* Second program *)
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 *)

Prog

(Python)
from operator import mul
from functools import reduce
def f(n): return n*(n + 1)//2
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)])
print([v(n) for n in range(1, 11)]) # Indranil Ghosh, Jul 24 2017
(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
(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

Crossrefs

Cf. A074962, A203310, A203467.

Sequence in context: A369681 A077521 A272445 * A371205 A090600 A113511

Adjacent sequences: A203306 A203307 A203308 * A203310 A203311 A203312

Keyword

nonn

Author

Clark Kimberling, Jan 01 2012

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 29 2023

Status

approved