A348084 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - k * x).
1, 3, 65, 2646, 159027, 12662650, 1256328866, 149304004500, 20677182465555, 3270191625210510, 581535955088511150, 114860168436414644100, 24947615188488584876910, 5909902157669174347277556, 1516413100622001261250104100, 418965782063742792530650053000
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..300
Programs
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PARI
a(n) = polcoef(1/prod(k=1, 2*n, 1-k*x+x*O(x^n)), n);
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PARI
a(n) = stirling(3*n, 2*n, 2);
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PARI
a(n) = sum(k=0, 2*n, (-1)^k*k^(3*n)*binomial(2*n, k))/(2*n)!;
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Python
from sympy.functions.combinatorial.numbers import stirling def A348084(n): return stirling(3*n,2*n) # Chai Wah Wu, Jun 09 2025
Formula
a(n) = S2(3*n,2*n), where S2(n,k) is Stirling numbers of the second kind.
a(n) = (1/(2*n)!) * Sum_{k=0..2*n} (-1)^k * k^(3*n) * binomial(2*n,k).
a(n) ~ 3^(3*n) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * 2^(n+1) * exp(n) * (3 - 2*c)^n * c^(2*n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292093838... - Vaclav Kotesovec, Oct 02 2021