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Revision History for A203428

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A203428

Reciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).
(history; published version)

#10 by OEIS Server at Thu Sep 28 02:03:45 EDT 2023

LINKS

G. C. Greubel, <a href="/A203428/b203428_1.txt">Table of n, a(n) for n = 1..33</a>

#9 by Joerg Arndt at Thu Sep 28 02:03:45 EDT 2023

STATUS

reviewed

approved

Discussion

Thu Sep 28

02:03

OEIS Server: Installed first b-file as b203428.txt.

#8 by Michel Marcus at Thu Sep 28 01:10:21 EDT 2023

STATUS

proposed

reviewed

#7 by G. C. Greubel at Thu Sep 28 01:01:49 EDT 2023

STATUS

editing

proposed

#6 by G. C. Greubel at Thu Sep 28 01:01:40 EDT 2023

LINKS

G. C. Greubel, <a href="/A203428/b203428_1.txt">Table of n, a(n) for n = 1..33</a>

FORMULA

a(n) = (-3)^binomial(n,2) * (Gamma(n+1))^(n-1) / BarnesG(n+1). - G. C. Greubel, Sep 28 2023

MATHEMATICA

(* First program *)

f[j_] := 1/(3 *j); z = 16;

v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]

1/Table[v[n], {n, 1, z}] (* A203428 *)

Table[v[n]/(3 *v[n + 1]), {n, 1, z - 1}] (* A203429 *)

(* Second program *)

Table[(-3)^Binomial[n, 2]*(Gamma[n+1])^(n-1)/BarnesG[n+1], {n, 20}] (* G. C. Greubel, Sep 28 2023 *)

PROG

(Magma)

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;

A203428:= func< n | (-3)^Binomial(n, 2)*(Factorial(n))^n/Barnes(n+1) >;

[A203428(n): n in [1..25]]; // G. C. Greubel, Sep 28 2023

(SageMath)

def barnes(n): return product(factorial(j) for j in range(n))

def A203428(n): return (-3)^binomial(n, 2)*(factorial(n))^n/barnes(n+1)

[A203428(n) for n in range(1, 21)] # G. C. Greubel, Sep 28 2023

CROSSREFS

Cf. A203429, A203424, A203421, A203424, A203429.

STATUS

approved

editing

#5 by Russ Cox at Fri Mar 30 18:58:06 EDT 2012

AUTHOR

_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 02 2012

Discussion

Fri Mar 30

18:58

OEIS Server: https://oeis.org/edit/global/285

#4 by T. D. Noe at Tue Jan 03 18:44:41 EST 2012

STATUS

proposed

approved

#3 by Clark Kimberling at Tue Jan 03 10:45:16 EST 2012

STATUS

editing

proposed

#2 by Clark Kimberling at Mon Jan 02 09:47:46 EST 2012

NAME

allocated for Clark KimberlingReciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).

DATA

1, -6, -486, 839808, 42515280000, -80335512599040000, -6890065294166289123840000, 31601087581187838970614157148160000, 8925080517850366815864624583251321642024960000

OFFSET

1,2

COMMENTS

Each term divides its successor, as in A203429.

MATHEMATICA

f[j_] := 1/(3 j); z = 16;

v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]

1/Table[v[n], {n, 1, z}] (* A203428 *)

Table[v[n]/(3 v[n + 1]), {n, 1, z - 1}] (* A203429 *)

CROSSREFS

Cf. A203429, A203424, A203421.

KEYWORD

allocated

sign

AUTHOR

Clark Kimberling (ck6(AT)evansville.edu), Jan 02 2012

STATUS

approved

editing

#1 by Clark Kimberling at Sun Jan 01 18:37:23 EST 2012

NAME

allocated for Clark Kimberling

KEYWORD

allocated

STATUS

approved