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A266499 Number of partitions of n with product of multiplicities of parts equal to n.

%I A266499 #27 Dec 22 2016 12:03:34
%S A266499 0,1,1,1,1,1,1,1,2,2,1,1,2,1,2,2,8,1,5,1,11,6,5,1,48,7,9,21,39,1,104,
%T A266499 1,143,27,20,45,457,1,32,58,620,1,549,1,363,514,65,1,4302,118,858,207,
%U A266499 926,1,4080,437,5171,382,181,1,20398,1,251,4287,20582,1212
%N A266499 Number of partitions of n with product of multiplicities of parts equal to n.
%H A266499 Alois P. Heinz, <a href="/A266499/b266499.txt">Table of n, a(n) for n = 0..1000</a>
%F A266499 a(n) = A266477(n,n).
%F A266499 p in primes => a(p) = 1.
%e A266499 a(8) = 2 because among the 22 (= A000041(8)) partitions of 8 only [1,1,1,1,1,1,1,1] and [1,1,1,1,2,2] have product of multiplicities of parts equal to 8.
%p A266499 b:= proc(n, i, p) option remember; `if`(p=1 and i*(i+1)/2<n, 0,
%p A266499      `if`(n=0, `if`(p=1, 1, 0), `if`(i<1, 0, b(n, i-1, p)+add(
%p A266499      `if`(irem(p, j)=0, b(n-i*j, i-1, p/j), 0), j=1..min(p, n/i)))))
%p A266499     end:
%p A266499 a:= n-> `if`(isprime(n), 1, b(n$3)):
%p A266499 seq(a(n), n=0..70);
%t A266499 b[n_, i_, p_] := b[n, i, p] = If[p == 1 && i*(i + 1)/2 < n, 0, If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, Min[p, n/i]}]]]]; a[n_] := If[PrimeQ[n], 1, b[n, n, n]]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A266499 Main diagonal of A266477.
%Y A266499 Cf. A000040, A000041.
%K A266499 nonn
%O A266499 0,9
%A A266499 _Emeric Deutsch_ and _Alois P. Heinz_, Dec 30 2015