A365381 - OEIS

A365381

Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 5, 2, 2, 1, 16, 8, 8, 10, 10, 7, 5, 5, 2, 2, 1, 32, 16, 16, 20, 20, 23, 15, 15, 12, 12, 8, 5, 5, 2, 2, 1, 64, 32, 32, 40, 40, 46, 47, 38, 33, 35, 29, 28, 21, 17, 14, 13, 8, 5, 5, 2, 2, 1, 128, 64, 64, 80, 80, 92, 94, 102, 79, 82, 76, 75, 68, 64, 53, 48, 43, 34, 33, 23, 19, 15, 13, 8, 5, 5, 2, 2, 1

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OFFSET

0,2

COMMENTS

Row lengths are A000124(n) = 1 + n*(n+1)/2.

LINKS

Table of n, a(n) for n=0..91.

EXAMPLE

Triangle begins:

2 1

4 2 2 1

8 4 4 5 2 2 1

16 8 8 10 10 7 5 5 2 2 1

32 16 16 20 20 23 15 15 12 12 8 5 5 2 2 1

64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13 8 5 5 2 2 1

Array begins:

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9

-------------------------------------------------------

n=0: 1

n=1: 2 1

n=2: 4 2 2 1

n=3: 8 4 4 5 2 2 1

n=4: 16 8 8 10 10 7 5 5 2 2

n=5: 32 16 16 20 20 23 15 15 12 12

n=6: 64 32 32 40 40 46 47 38 33 35

n=7: 128 64 64 80 80 92 94 102 79 82

n=8: 256 128 128 160 160 184 188 204 207 184

n=9: 512 256 256 320 320 368 376 408 414 440

The T(5,8) = 12 subsets are:

{3,5} {1,2,5} {1,2,3,4} {1,2,3,4,5}

{1,3,4} {1,2,3,5}

{1,3,5} {1,2,4,5}

{2,3,5} {1,3,4,5}

{3,4,5} {2,3,4,5}

MATHEMATICA

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 8}, {k, 0, n*(n+1)/2}]

CROSSREFS

Row lengths are A000124 = number of distinct sums of subsets of {1..n}.

Central column/main diagonal is A365376.

A000009 counts sets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A365046 counts combination-full subsets, differences of A364914.

Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.

Sequence in context: A323915 A080100 A365425 * A161822 A001176 A136693

Adjacent sequences: A365378 A365379 A365380 * A365382 A365383 A365384

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Sep 08 2023

STATUS

approved