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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].

%I A375250 #15 Aug 10 2024 09:16:33
%S A375250 1,2,6,2,30,12,42,6,30,20,44,12,910,420,30,6,102,12,7980,420,13860,
%T A375250 1320,4140,180,2730,1092,84,28,58,60,2046,66,117810,7140,420,12,36556,
%U A375250 9880,780,20,189420,9240,397320,9240,48300,19320,19740,1260,46410,39780,87516,1716,6996,264
%N 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].
%H A375250 J. S. Dowker, <a href="https://doi.org/10.48550/arXiv.1108.1760">Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves</a>, arXiv:1108.1760 [math.NT], 2011. (See the factor in formula (27).)
%H A375250 G. J. Rieger, <a href="https://eudml.org/doc/160721">Über Partitionen</a>, Mathematische Annalen (1959), Volume: 138, page 356-362. (See Satz 1.)
%H A375250 J. J. Sylvester, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.$b417523&amp;seq=161">On the partition of numbers</a>, Quarterly J. Pure Appl. Math. 1857, 1:141-152.
%F A375250 a(n) = denominator(W(n))/(n!*(n - 1)!) where W(n) = [t^(-1)] exp(t*x)/ Product_{k=1..n}(1 - exp(-t*k)).
%p A375250 read(PARTITIONS):  # From the paper of Sills & Zeilberger cited in A375252.
%p A375250 a := n -> denom(op(pmnPC(n, x)[1])) / (n!*(n - 1)!):
%p A375250 seq(a(n), n = 1..54);
%p A375250 # Or, standalone:
%p A375250 W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);
%p A375250 expand(series(%, t, n+1)); coeff(%, t, -1) end:
%p A375250 a := n -> n*denom(W(n))/(n!^2): seq(a(n), n = 1..24);
%Y A375250 Cf. A375251, A375252, A010790, A008284.
%K A375250 nonn
%O A375250 1,2
%A A375250 _Peter Luschny_, Aug 09 2024