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

Author

Mats Granvik, May 16 2010

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

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

Cf. A177975, A177977, A344527, A345229.

Programs

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

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)