login
Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.
9

%I #9 Feb 23 2024 17:24:36

%S 1,2,2,8,8,32,144,432,2160,27000,582120,7623000,336936600,6740402760,

%T 543454231320,57619849046760,4683793138766280,412882704970215480,

%U 88171665744392750520,12780536107937124847320,2685589660883755945879560,942036670625665177379096280

%N Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

%C Also the Heinz number of row n of A015716 (with zeros removed).

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%H Alois P. Heinz, <a href="/A325513/b325513.txt">Table of n, a(n) for n = 0..172</a>

%F a(n) = A181819(A003963(A325505(n))).

%F A056239(a(n)) = A015723(n).

%e The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 2: {1}

%e 8: {1,1,1}

%e 8: {1,1,1}

%e 32: {1,1,1,1,1}

%e 144: {1,1,1,1,2,2}

%e 432: {1,1,1,1,2,2,2}

%e 2160: {1,1,1,1,2,2,2,3}

%e 27000: {1,1,1,2,2,2,3,3,3}

%e 582120: {1,1,1,2,2,2,3,4,4,5}

%e 7623000: {1,1,1,2,2,3,3,3,4,5,5}

%e 336936600: {1,1,1,2,2,3,3,4,5,5,6,7}

%e 6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}

%e 543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}

%e 57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

%p b:= proc(n, i) option remember;

%p `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+

%p (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))

%p end:

%p a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(

%p coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):

%p seq(a(n), n=0..21); # _Alois P. Heinz_, Feb 23 2024

%t Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

%Y Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

%K nonn

%O 0,2

%A _Gus Wiseman_, May 07 2019