A177975 - OEIS

A177975

Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 7, 9, 4, 1, 0, 2, 14, 16, 14, 5, 1, 0, 6, 13, 34, 30, 20, 6, 1, 0, 4, 27, 43, 69, 50, 27, 7, 1, 0, 6, 26, 83, 107, 125, 77, 35, 8, 1, 0, 4, 39, 100, 209, 226, 209, 112, 44, 9, 1, 0, 10, 38, 155, 295, 461, 428, 329, 156, 54, 10, 1

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OFFSET

1,8

COMMENTS

The first column in this array is the first column in A134540 which is the matrix inverse of A177978. The second column is the first column in A159905. The rows in this array are described by both binomial expressions and closed form polynomials.

LINKS

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

FORMULA

From Seiichi Manyama, Jun 12 2021: (Start)

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

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

EXAMPLE

Table begins:

1..1...1...1....1.....1.....1......1......1.......1.......1

0..1...2...3....4.....5.....6......7......8.......9......10

0..2...5...9...14....20....27.....35.....44......54......65

0..2...7..16...30....50....77....112....156.....210.....275

0..4..14..34...69...125...209....329....494.....714....1000

0..2..13..43..107...226...428....749...1234....1938....2927

0..6..27..83..209...461...923...1715...3002....5004....8007

0..4..26.100..295...736..1632...3312...6270...11220...19162

0..6..39.155..480..1266..2975...6399..12825...24255...43692

0..4..38.182..641..1871..4789..11103..23807...47896...91367

0.10..65.285.1000..3002..8007..19447..43757...92377..184755

0..4..50.292.1209..4066.11837..30920..74139..165748..349438

0.12..90.454.1819..6187.18563..50387.125969..293929..646645

0..6..75.473.2166..8101.26202..75797.200479..492406.1136048

0..8.100.636.2976.11482.38523.115915.319231..816421.1960190

0..8.100.696.3546.14712.52548.167112.483879.1296064.3249312

PROG

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

CROSSREFS

Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.

Main diagonal gives A332470.

Cf. A177976, A177977.

Sequence in context: A280817 A060086 A308680 * A340995 A363733 A062135

Adjacent sequences: A177972 A177973 A177974 * A177976 A177977 A177978

KEYWORD

nonn,tabl

AUTHOR

Mats Granvik, May 16 2010

STATUS

approved