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A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

%I A304442 #36 Jun 26 2022 14:03:25
%S A304442 1,1,2,2,4,2,5,2,7,3,5,2,13,2,5,4,11,2,13,2,12,4,5,2,28,3,5,5,12,2,18,
%T A304442 2,17,4,5,4,44,2,5,4,24,2,18,2,12,10,5,2,63,3,9,4,12,2,34,4,24,4,5,2,
%U A304442 67,2,5,10,27,4,18,2,12,4,14,2,120,2,5,7,12,4,18,2,54
%N A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.
%C A304442 Said differently, these are partitions whose run-sums are all equal. - _Gus Wiseman_, Jun 25 2022
%H A304442 Seiichi Manyama, <a href="/A304442/b304442.txt">Table of n, a(n) for n = 0..10000</a>
%F A304442 a(n) >= 2 for n > 1.
%F A304442 a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.
%e A304442 a(72) = binomial(d(72),1) + binomial(d(36),2) + binomial(d(24),3) + binomial(d(18),4) + binomial(d(12),6) = 12 + 36 + 56 + 15 + 1 = 120, where d(n) is the number of divisors of n.
%e A304442 --+----------------------+-----------------------------------------
%e A304442 n |                      | Sequence of the sum of the same summands
%e A304442 --+----------------------+-----------------------------------------
%e A304442 1 | 1                    | 1
%e A304442 2 | 2                    | 2
%e A304442   | 1+1                  | 2
%e A304442 3 | 3                    | 3
%e A304442   | 1+1+1                | 3
%e A304442 4 | 4                    | 4
%e A304442   | 2+2                  | 4
%e A304442   | 2+1+1                | 2, 2
%e A304442   | 1+1+1+1              | 4
%e A304442 5 | 5                    | 5
%e A304442   | 1+1+1+1+1            | 5
%e A304442 6 | 6                    | 6
%e A304442   | 3+3                  | 6
%e A304442   | 3+1+1+1              | 3, 3
%e A304442   | 2+2+2                | 6
%e A304442   | 1+1+1+1+1+1          | 6
%t A304442 Table[Length[Select[IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}] (* _Gus Wiseman_, Jun 25 2022 *)
%o A304442 (PARI) a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ _Michel Marcus_, May 13 2018
%Y A304442 Cf. A000005 (d(n)), A304405, A304406, A304428, A304430.
%Y A304442 All parts are divisors of n, see A018818, compositions A100346.
%Y A304442 For run-lengths instead of run-sums we have A047966, compositions A329738.
%Y A304442 These partitions are ranked by A353833.
%Y A304442 The distinct instead of equal version is A353837, ranked by A353838, compositions A353850.
%Y A304442 The version for compositions is A353851, ranked by A353848.
%Y A304442 Cf. A098504, A098859, A275870, A353832, A353847, A353864, A353932.
%K A304442 nonn
%O A304442 0,3
%A A304442 _Seiichi Manyama_, May 12 2018