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A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.

%I A303974 #10 Sep 18 2018 16:59:24
%S A303974 1,2,1,3,3,3,4,6,10,6,5,10,22,23,15,6,15,40,57,62,27,7,21,65,115,165,
%T A303974 129,63,8,28,98,205,356,385,318,120,9,36,140,336,676,914,1005,676,252,
%U A303974 10,45,192,518,1176,1885,2524,2334,1524,495,11,55,255,762,1918,3528,5495,6319,5607,3261,1023
%N A303974 Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.
%C A303974 A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.
%H A303974 Andrew Howroyd, <a href="/A303974/b303974.txt">Table of n, a(n) for n = 1..1275</a>
%F A303974 T(n,k) = Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - _Andrew Howroyd_, Sep 18 2018
%e A303974 Triangle begins:
%e A303974 1
%e A303974 2    1
%e A303974 3    3    3
%e A303974 4    6   10    6
%e A303974 5   10   22   23   15
%e A303974 6   15   40   57   62   27
%e A303974 7   21   65  115  165  129   63
%e A303974 8   28   98  205  356  385  318  120
%e A303974 9   36  140  336  676  914 1005  676  252
%e A303974 The a(4,3) = 10 multisets: (112), (113), (122), (123), (124), (133), (134), (223), (233), (234).
%e A303974 The a(5,4) = 23 multisets:
%e A303974 (1112), (1222),
%e A303974 (1113), (1123), (1223), (1233), (1333), (2223), (2333),
%e A303974 (1124), (1134), (1224), (1234), (1244), (1334), (1344), (2234), (2334), (2344),
%e A303974 (1235), (1245), (1345), (2345).
%t A303974 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A303974 Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Length],{n,10}]
%o A303974 (PARI) T(n,k)={sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i)))} \\ _Andrew Howroyd_, Sep 18 2018
%Y A303974 Row sums are A303976.
%Y A303974 Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.
%K A303974 nonn,tabl
%O A303974 1,2
%A A303974 _Gus Wiseman_, May 03 2018
%E A303974 Terms a(56) and beyond from _Andrew Howroyd_, Sep 18 2018