A375251 - OEIS

%I #11 Aug 10 2024 09:02:31

%S 1,4,72,288,86400,1036800,152409600,1219276800,438939648000,

%T 26336378880000,6373403688960000,229442532802560000,

%U 2714305163054284800000,228001633696559923200000,3420024505448398848000000,164161176261523144704000000,759081279033283021111296000000

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

%F (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].

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

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

%p read(PARTITIONS):  # See the Sills & Zeilberger paper cited in A375252.

%p seq(denom(op(pmnPC(n,x)[1])), n = 1..17);

%p # Or, standalone:

%p W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);

%p expand(series(%, t, n+1)); coeff(%, t, -1) end:

%p a := n -> denom(W(n)): seq(a(n), n = 1..17);

%Y Cf. A375252 (numerators), A375250.

%K nonn,frac

%O 1,2

%A _Peter Luschny_, Aug 07 2024