A375251 Denominators of the polynomials A375252 (polynomial part of the partition function restricted to partitions of the integer x with parts in (1,2,...,n)).

1, 4, 72, 288, 86400, 1036800, 152409600, 1219276800, 438939648000, 26336378880000, 6373403688960000, 229442532802560000, 2714305163054284800000, 228001633696559923200000, 3420024505448398848000000, 164161176261523144704000000, 759081279033283021111296000000

Offset: 1

Author

Peter Luschny, Aug 07 2024

Keywords

• nonn
• frac

Crossrefs

Cf. A375252 (numerators), A375250.

Programs

Maple

read(PARTITIONS):  # See the Sills & Zeilberger paper cited in A375252.
seq(denom(op(pmnPC(n,x)[1])), n = 1..17);
# 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 -> denom(W(n)): seq(a(n), n = 1..17);

Formula

(Sum_{k=0..n-1} A375252(n, k)*x^k) / a(n) = W1([n], x), where W1([n], x) denotes the first Sylvester wave restricted to parts in [n].

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

a(n) = A375250(n)*n!*(n - 1)!.