A381056 Product of row n of A329708.
1, 16, 4320, 7680000, 56672000000, 1315328716800000, 79725223359774720000, 11041460968683995136000000, 3159164253667495772160000000000, 1725992749819407775039488000000000000, 1690274868390850110509130354524160000000000, 2816890048270042497343000411961733572198400000000
Offset: 0
Keywords
Examples
Row n=2 of A329708 is {1, 4, 10, 12, 9} and the product of those is a(2) = 4320.
Programs
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PARI
a(n) = vecprod(Vec(sum(k=0, n, (k+1)*x^k)^2)); \\ Michel Marcus, Feb 13 2025
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Python
from sympy import prod def a(n): p = ((n+1)*(n+2)*(n+3)) // 6 p *= prod(((k*(k+1)*(k+2))*((n+k+1)*(n+k+2)*(n+k+3)-2*k*(k+1)*(3*n+k+5)))//36 for k in range(1,n+1)) return p print([a(n) for n in range(0, 14)])
Formula
a(n) = Product_{k=0..2*n} A329708(n,k).
a(n) = Product_{k=0..2*n} (Sum_{i=max(0,k-n)..min(k,n)} (i+1)*(k-i+1)).
a(n) == 0 (mod (n+1)^3).
a(n) = (n+1)*(n+2)*(n+3)*(1/6)*(Product_{k=1..n} k*(k+1)*(k+2)*((n+k+1)*(n+k+2)*(n+k+3)-2*k*(k+1)*(3*n+k+5))/36).
a(n) = binomial(n+3,3)*(Product_{k=1..n} binomial(k+2,3)*(binomial(k+n+3,3)-(k*(k+1)*(3*n+k+5))/3)).
a(n) = (Product_{k=0..n-1} Sum_{i=0..k} (i+1)*(k-i+1)) * (Product_{k=n..2*n} Sum_{i=k-n..n} (i+1)*(k-i+1)).
a(n) = (Product_{k=0..n-1} binomial(k+3,3))*(Product{k=n..2*n} binomial(k+3,3)-(2*n+4)*binomial(k+2-n)+(2*n+2)*binomial(k+1-n)).