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%I #9 Sep 10 2023 01:42:22
%S 1,1,2,1,6,2,24,1,4,6,120,2,720,24,12,1,5040,4,40320,6,48,120,362880,
%T 2,36,720,8,24,3628800,12,39916800,1,240,5040,144,4,479001600,40320,
%U 1440,6,6227020800,48,87178291200,120,24,362880,1307674368000,2,576,36,10080
%N Sum of coefficients of power sums symmetric functions in h(y) * Product_i y_i! where h is homogeneous symmetric functions and y is the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>.
%F Totally multiplicative with a(p) = primepi(p)! = A000142(A000720(p)). - _Amiram Eldar_, Sep 10 2023
%e The sum of coefficients of 12h(32) = 2p(32) + 3p(221) + 2p(311) + 4p(2111) + p(11111) is a(15) = 12.
%t f[p_, e_] := (PrimePi[p]!)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Sep 10 2023 *)
%Y Row sums of A321897.
%Y Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A321742-A321765.
%Y Cf. A000142, A000720.
%K nonn,easy,mult
%O 1,3
%A _Gus Wiseman_, Nov 20 2018