A177976 Square array T(n,k) read by antidiagonals up. Cumulative column sums of A177975.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 10, 15, 13, 5, 1, 1, 12, 29, 29, 19, 6, 1, 1, 18, 42, 63, 49, 26, 7, 1, 1, 22, 69, 106, 118, 76, 34, 8, 1, 1, 28, 95, 189, 225, 201, 111, 43, 9, 1, 1, 32, 134, 289, 434, 427, 320, 155, 53, 10, 1, 1, 42, 172, 444, 729, 888, 748, 484, 209, 64, 11, 1

Offset

1,5

Comments

Each row is described by both a binomial expression and a closed form polynomial. The closed form polynomials given in A177977 extends this table to the left. For example the 0th column is A002321 and the -1st column is A092149.

Also number of ordered k-tuples of integers from [ 1..n ] with no global factor. - Seiichi Manyama, Jun 12 2021

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Formula

From Seiichi Manyama, Jun 12 2021: (Start)

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} mu(j) * x^j/(1 - x^j)^k.

T(n,k) = Sum_{j=1..n} Sum_{d|j} mu(j/d) * binomial(d+k-2,d-1).

T(n,k) = binomial(n+k-1,k) - Sum_{j=2..n} T(floor(n/j),k). (End)

Example

Table begins:
  1..1...1....1.....1.....1......1......1.......1.......1.......1
  1..2...3....4.....5.....6......7......8.......9......10......11
  1..4...8...13....19....26.....34.....43......53......64......76
  1..6..15...29....49....76....111....155.....209.....274.....351
  1.10..29...63...118...201....320....484.....703.....988....1351
  1.12..42..106...225...427....748...1233....1937....2926....4278
  1.18..69..189...434...888...1671...2948....4939....7930...12285
  1.22..95..289...729..1624...3303...6260...11209...19150...31447
  1.28.134..444..1209..2890...6278..12659...24034...43405...75139
  1.32.172..626..1850..4761..11067..23762...47841...91301..166506
  1.42.237..911..2850..7763..19074..43209...91598..183678..351261
  1.46.287.1203..4059.11829..30911..74129..165737..349426..700699
  1.58.377.1657..5878.18016..49474.124516..291706..643355.1347344
  1.64.452.2130..8044.26117..75676.200313..492185.1135761.2483392
  1.72.552.2766.11020.37599.114199.316228..811416.1952182.4443582
  1.80.652.3462.14566.52311.166747.483340.1295295.3248246.7692894

Prog

(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, moebius(j/d)*binomial(d+k-2, d-1))); \\ Seiichi Manyama, Jun 12 2021
(PARI) T(n, k) = binomial(n+k-1, k)-sum(j=2, n, T(n\j, k)); \\ Seiichi Manyama, Jun 12 2021

Crossrefs

Column k=1..5 gives A000012, A002088, A015631, A015634, A015650.

Cf. A177975, A177977, A344527, A345229.

Sequence in context: A230858 A060098 A161492 * A034781 A110470 A347699

Adjacent sequences: A177973 A177974 A177975 * A177977 A177978 A177979

Keyword

nonn,tabl

Author

Mats Granvik, May 16 2010

Status

approved