A203310 -id:A203310 - cp's OEIS frontend

A203309 Vandermonde determinant of the first n triangular numbers.

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

Offset: 0

Clark Kimberling, Jan 01 2012

nonn

Each term divides its successor, as in A203310.

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

Cf. A074962, A203310, A203467.

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

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

(* 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 *)

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

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

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)

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

A203467 a(n) = A203309(n)/A000178(n) where A000178 are superfactorials.

Original entry on oeis.org

1, 1, 2, 15, 630, 198450, 589396500, 19912024006875, 8969371213896843750, 61815874928487448987968750, 7358663747680777931818630148437500, 16862758880642741957030086746987589746093750

Offset: 0

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..39
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A203309, A203310.

F:= Factorial; [1] cat [(&*[(F(2*k+2))/(2^k*F(k+2)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,0,z}]           (* A203309 *)
Table[v[n+1]/v[n], {n,z}]      (* A203310 *)
Table[v[n]/d[n], {n,0,12}]     (* A203467 *)
(* Second program *)
Table[Product[(2*k+2)!/(2^k*(k+2)!), {k,n-1}], {n,0,20}] (* G. C. Greubel, Aug 29 2023 *)

f=factorial; [product((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

From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).
a(n) = (1/2)^binomial(n,2)*BarnesG(n+1)*Product_{k=2..n} binomial(2*k, k+1).
a(n) = Product_{k=1..n-1} (2*k+2)!/(2^k*(k+2)!). (End)
a(n) ~ sqrt(A/Pi) * 2^(n^2/2 + 2*n - 7/24) * n^(n^2/2 - n/2 - 35/24) / exp(3*n^2/4 - n/2 + 1/24), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023

Name edited by Michel Marcus, May 17 2019

a(0) = 1 prepended by G. C. Greubel, Aug 29 2023

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A203309 Vandermonde determinant of the first n triangular numbers.

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