A376524 -id:A376524 - cp's OEIS frontend

A376523 a(n) = Product_{k=0..n} (k^3 + n - k).

0, 1, 32, 2187, 286720, 64796875, 23279477760, 12506434235113, 9582123576983552, 10084099499408154825, 14139206937856000000000, 25756714724499975610869475, 59683270195198565091221962752, 172781591936242461223503558613507, 615312169743368293769528795463680000

Offset: 0

Vaclav Kotesovec, Sep 26 2024

nonn

Cf. A323541, A374881, A374886, A376524.

A376523 := proc(n)
    mul(k^3+n-k,k=0..n) ;
end proc:
seq(A376523(n),n=0..20) ; # R. J. Mathar, Sep 27 2024

Table[Product[k^3+n-k, {k, 0, n}], {n, 0, 16}]

a(n) ~ exp(2*Pi*n^(1/3)/sqrt(3) - 3*n) * n^(3*n+2) * (1 - 2*Pi/(3^(3/2)*n^(1/3)) + 2*Pi^2/(27*n^(2/3)) + (27/40 - 4*Pi^3/(243*sqrt(3)))/n).

A374881 Obverse convolution (n)**(n^2); see Comments.

Original entry on oeis.org

0, 1, 16, 405, 15360, 818125, 58226688, 5332085577, 610140160000, 85235284359225, 14264819712000000, 2815701027697558429, 646960843646287478784, 171112492588968115453125, 51595090958399913852928000, 17587698619968027952119140625

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000027, A000290, A374848, A376523, A376524.

s[n_] := n; t[n_] := n^2;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 18}]

a(n) ~ n^(2*n + 1) / exp(2*n + 1 - Pi*sqrt(n)). - Vaclav Kotesovec, Jul 31 2024

cp's OEIS Frontend

A376523 a(n) = Product_{k=0..n} (k^3 + n - k).

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