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