A203431 -id:A203431 - cp's OEIS frontend

A203430 Vandermonde determinant of the first n numbers (1,3,4,6,7,9,10,...) = (j+floor(j/2)).

1, 2, 6, 180, 12960, 18662400, 84652646400, 12068081270784000, 6568897997313146880000, 157325632547489652827750400000, 16698920220108665726304214056960000000, 101984821172231138973752227905335721984000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203431, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203432.

G. C. Greubel, Table of n, a(n) for n = 1..40

Cf. A032766, A203431, A203432.

A203430:= func< n | n eq 1 select 1 else (&*[(&*[k-j+Floor((k+1)/2)-Floor((j+1)/2): j in [0..k-1]]) : k in [1..n-1]]) >;
[A203430(n): n in [1..25]]; // G. C. Greubel, Sep 27 2023

f[j_]:= j + Floor[j/2]; z = 20;
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,z}]            (* this sequence *)
Table[v[n+1]/v[n], {n,z}]     (* A203431 *)
Table[v[n]/d[n], {n,z}]       (* A203432 *)

def A203430(n): return product(product(k-j+((k+1)//2)-((j+1)//2) for j in range(k)) for k in range(1, n))
[A203430(n) for n in range(1, 31)] # G. C. Greubel, Sep 27 2023

A203432 a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).

Original entry on oeis.org

1, 2, 3, 15, 45, 540, 3402, 96228, 1299078, 85739148, 2507870079, 383704122087, 24487299427734, 8645900336407620, 1209640056157393380, 982320774834892454820, 302358334494179897593596, 563293577162657149216869348

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..100
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A032766, A203430, A203431.

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
A203432:= func< n | n eq 1 select 1 else (&*[(&*[k-j+Floor((k+1)/2)-Floor((j+1)/2): j in [0..k-1]]) : k in [1..n-1]])/Barnes(n) >;
[A203432(n): n in [1..25]]; // G. C. Greubel, Sep 20 2023

f[j_]:= j + Floor[j/2]; z = 20;
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,z}]             (* A203430 *)
Table[v[n+1]/v[n], {n,z}]      (* A203431 *)
Table[v[n]/d[n], {n,z}]        (* this sequence *)

def barnes(n): return product(factorial(j) for j in range(n))
def A203432(n): return product(product(k-j+((k+1)//2)-((j+1)//2) for j in range(k)) for k in range(1,n))/barnes(n)
[A203432(n) for n in range(1,31)] # G. C. Greubel, Sep 20 2023

cp's OEIS Frontend

A203430 Vandermonde determinant of the first n numbers (1,3,4,6,7,9,10,...) = (j+floor(j/2)).

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