A013580 Triangle formed in same way as Pascal's triangle (A007318) except 1 is added to central element in even-numbered rows.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 9, 5, 1, 1, 6, 14, 14, 6, 1, 1, 7, 20, 29, 20, 7, 1, 1, 8, 27, 49, 49, 27, 8, 1, 1, 9, 35, 76, 99, 76, 35, 9, 1, 1, 10, 44, 111, 175, 175, 111, 44, 10, 1, 1, 11, 54, 155, 286, 351, 286, 155, 54, 11, 1, 1, 12, 65, 209, 441, 637, 637, 441, 209, 65

Offset

0,5

Comments

From Gus Wiseman, Apr 19 2023: (Start)

Appears to be the number of nonempty subsets of {1,...,n} with median k, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:

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

{1,3} {1,5} {3,5}

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

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

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

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Including half-steps gives A231147.

For mean instead of median we have A327481.

(End)

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

G.f.: 1/(1-(1+y)*x)/(1-y*x^2). - Vladeta Jovovic, Oct 12 2003

Example

Triangle begins:
   1
   1   1
   1   3   1
   1   4   4   1
   1   5   9   5   1
   1   6  14  14   6   1
   1   7  20  29  20   7   1
   1   8  27  49  49  27   8   1
   1   9  35  76  99  76  35   9   1
   1  10  44 111 175 175 111  44  10   1
   1  11  54 155 286 351 286 155  54  11   1
   1  12  65 209 441 637 637 441 209  65  12   1

Mathematica

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x)/(1 - y*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 10 2017 *)

Crossrefs

Row sums give A000975, A054106.

Central diagonal T(2n+1,n+1) appears to be A006134.

Central diagonal T(2n,n) appears to be A079309.

For partitions instead of subsets we have A359901, row sums A325347.

A000975 counts subsets with integer median.

A007318 counts subsets by length, A359893 by twice median.

Cf. A000984, A024718, A057552, A231147, A327475, A327481, A361654.

Sequence in context: A028262 A173117 A050177 * A147290 A026670 A131402

Adjacent sequences: A013577 A013578 A013579 * A013581 A013582 A013583

Keyword

tabl,nonn,easy

Author

Martin Hecko (bigusm(AT)interramp.com)

Extensions

More terms from James A. Sellers

Status

approved