A375250 a(n) = A375251(n) / A010790(n) = denominator(W1([n], x)) / (n!*(n - 1)!), where W1([n], x) is the first Sylvester wave for parts in [n].

1, 2, 6, 2, 30, 12, 42, 6, 30, 20, 44, 12, 910, 420, 30, 6, 102, 12, 7980, 420, 13860, 1320, 4140, 180, 2730, 1092, 84, 28, 58, 60, 2046, 66, 117810, 7140, 420, 12, 36556, 9880, 780, 20, 189420, 9240, 397320, 9240, 48300, 19320, 19740, 1260, 46410, 39780, 87516, 1716, 6996, 264

Offset: 1

Author

Peter Luschny, Aug 09 2024

Keywords

• nonn

Links

• J. S. Dowker, Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves, arXiv:1108.1760 [math.NT], 2011. (See the factor in formula (27).)
• G. J. Rieger, Über Partitionen, Mathematische Annalen (1959), Volume: 138, page 356-362. (See Satz 1.)
• J. J. Sylvester, On the partition of numbers, Quarterly J. Pure Appl. Math. 1857, 1:141-152.

Crossrefs

Cf. A375251, A375252, A010790, A008284.

Programs

Maple

read(PARTITIONS):  # From the paper of Sills & Zeilberger cited in A375252.
a := n -> denom(op(pmnPC(n, x)[1])) / (n!*(n - 1)!):
seq(a(n), n = 1..54);
# Or, standalone:
W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);
expand(series(%, t, n+1)); coeff(%, t, -1) end:
a := n -> n*denom(W(n))/(n!^2): seq(a(n), n = 1..24);

Formula

a(n) = denominator(W(n))/(n!*(n - 1)!) where W(n) = [t^(-1)] exp(t*x)/ Product_{k=1..n}(1 - exp(-t*k)).