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, 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

Crossrefs

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

Sequence in context: A136600 A136672 A097750 * A133544 A303872 A013609

Adjacent sequences: A304620 A304621 A304622 * A304624 A304625 A304626

Keyword

nonn,tabl

Author

Gus Wiseman, May 15 2018

Status

approved