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A177976
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Square array T(n,k) read by antidiagonals up. Cumulative column sums of A177975.
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7
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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