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A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

%I A304623 #20 Jan 21 2023 11:58:20
%S A304623 1,1,2,1,4,4,1,6,11,8,1,10,21,27,16,1,12,38,61,63,32,1,18,57,120,162,
%T A304623 143,64,1,22,87,205,347,409,319,128,1,28,122,333,651,950,1000,703,256,
%U A304623 1,32,164,506,1132,1926,2504,2391,1535,512,1,42,217,734,1840
%N A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.
%C A304623 A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.
%H A304623 Andrew Howroyd, <a href="/A304623/b304623.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A304623 T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - _Andrew Howroyd_, Jan 20 2023
%e A304623 Triangle begins:
%e A304623 1
%e A304623 1    2
%e A304623 1    4    4
%e A304623 1    6   11    8
%e A304623 1   10   21   27   16
%e A304623 1   12   38   61   63   32
%e A304623 1   18   57  120  162  143   64
%e A304623 1   22   87  205  347  409  319  128
%e A304623 The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).
%t A304623 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A304623 Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]
%o A304623 (PARI) T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ _Andrew Howroyd_, Jan 20 2023
%Y A304623 Row sums are A303976.
%Y A304623 Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.
%K A304623 nonn,tabl
%O A304623 1,3
%A A304623 _Gus Wiseman_, May 15 2018