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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)).

%I A375251 #11 Aug 10 2024 09:02:31
%S A375251 1,4,72,288,86400,1036800,152409600,1219276800,438939648000,
%T A375251 26336378880000,6373403688960000,229442532802560000,
%U A375251 2714305163054284800000,228001633696559923200000,3420024505448398848000000,164161176261523144704000000,759081279033283021111296000000
%N 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)).
%F A375251 (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 A375251 a(n) = denominator(W(n)) where W(n) = [t^(-1)] exp(t*x)/Product_{k=1..n}(1 - exp(-t*k)).
%F A375251 a(n) = A375250(n)*n!*(n - 1)!.
%p A375251 read(PARTITIONS):  # See the Sills & Zeilberger paper cited in A375252.
%p A375251 seq(denom(op(pmnPC(n,x)[1])), n = 1..17);
%p A375251 # Or, standalone:
%p A375251 W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);
%p A375251 expand(series(%, t, n+1)); coeff(%, t, -1) end:
%p A375251 a := n -> denom(W(n)): seq(a(n), n = 1..17);
%Y A375251 Cf. A375252 (numerators), A375250.
%K A375251 nonn,frac
%O A375251 1,2
%A A375251 _Peter Luschny_, Aug 07 2024