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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.
1
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, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840
OFFSET
1,3
COMMENTS
A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
FORMULA
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
EXAMPLE
Triangle begins:
1
1 2
1 4 4
1 6 11 8
1 10 21 27 16
1 12 38 61 63 32
1 18 57 120 162 143 64
1 22 87 205 347 409 319 128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).
MATHEMATICA
allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n], GCD@@Length/@Split[#]===1&], Max], {n, 10}]
PROG
(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
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 15 2018
STATUS
approved