A203430
Vandermonde determinant of the first n numbers (1,3,4,6,7,9,10,...) = (j+floor(j/2)).
Original entry on oeis.org
1, 2, 6, 180, 12960, 18662400, 84652646400, 12068081270784000, 6568897997313146880000, 157325632547489652827750400000, 16698920220108665726304214056960000000, 101984821172231138973752227905335721984000000000
Offset: 1
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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
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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 *)
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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
A203431
a(n) = v(n+1)/v(n), where v=A203430.
Original entry on oeis.org
2, 3, 30, 72, 1440, 4536, 142560, 544320, 23950080, 106142400, 6107270400, 30569011200, 2198617344000, 12197035468800, 1061932177152000, 6440034727526400, 662645678542848000, 4347023441080320000
Offset: 1
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A203431:= func< n | n eq 1 select 2 else (&*[n-j+Floor((n+1)/2)-Floor((j+1)/2): j in [0..n-1]]) >;
[A203431(n): n in [1..25]]; // G. C. Greubel, Sep 27 2023
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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}] (* this sequence *)
Table[v[n]/d[n], {n,z}] (* A203432 *)
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def A203431(n): return product(n-j+((n+1)//2)-((j+1)//2) for j in range(n))
[A203431(n) for n in range(1, 31)] # G. C. Greubel, Sep 27 2023
Original entry on oeis.org
1, 1, 3, 6, 45, 189, 3402, 30618, 1299078, 25332021, 2507870079, 106698472452, 24487299427734, 2283997201168644, 1209640056157393380, 248218139523497121576, 302358334494179897593596, 136861610819571430116630660
Offset: 1
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Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
[1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // G. C. Greubel, Sep 19 2023
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f[j_]:= j + Floor[(j+1)/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}] (* A203433 *)
Table[v[n+1]/v[n], {n,z}] (* A014402 *)
Table[v[n]/d[n], {n,z}] (* A203434 *)
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def barnes(n): return product(factorial(j) for j in range(n))
def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
[product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # G. C. Greubel, Sep 19 2023
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