A177978 -id:A177978 - cp's OEIS frontend

A177517 Triangle T(n,k) read by rows defined by recurrence T(n,1)=A000007(n-1) and T(n,k) = sum_{i=1..k-1} T(n-i,k-1) if k>1.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 1, 5, 4, 1, 0, 0, 0, 0, 6, 9, 5, 1, 0, 0, 0, 0, 5, 15, 14, 6, 1, 0, 0, 0, 0, 3, 20, 29, 20, 7, 1, 0, 0, 0, 0, 1, 22, 49, 49, 27, 8, 1, 0, 0, 0, 0, 0, 20, 71, 98, 76, 35, 9, 1, 0, 0, 0, 0, 0, 15, 90, 169, 174, 111, 44, 10, 1, 0, 0, 0, 0, 0, 9, 101, 259, 343, 285, 155, 54, 11, 1, 0, 0, 0, 0, 0, 4, 101, 359, 602, 628, 440, 209, 65, 12, 1, 0, 0, 0, 0, 0, 1, 90, 455, 961, 1230, 1068, 649, 274, 77, 13, 1

Offset: 1

Mats Granvik, Roger L. Bagula, Dec 11 2010

A008302 is the main entry for this triangle.
Essentially A060701 which is equal to this table beginning from the second column.
The recurrence formula is similar to the recurrence for A177978.

			1,
0,1,
0,0,1,
0,0,1,1,
0,0,0,2,1,
0,0,0,2,3,1,
0,0,0,1,5,4,1,
0,0,0,0,6,9,5,1,
0,0,0,0,5,15,14,6,1,
0,0,0,0,3,20,29,20,7,1,
0,0,0,0,1,22,49,49,27,8,1

Alois P. Heinz, Table of n, a(n) for n = 1..20301

Cf. A008302, A060701, A177978, A175105. Column sums are A000142. Row sums are A008930.

t[1, 1] = 1; t[n_, 1] = 0; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0];
Flatten[Table[t[n, k], {n, 12}, {k, n}]]
(* Robert G. Wilson v, Jun 24 2011 *) (* corrected by Mats Granvik, Jan 23 2012 *)

T(n,k) = A008302(k-2,n-k), n>=k>1. - R. J. Mathar, Dec 15 2010

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.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, May 16 2010

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.

			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

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

Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.
Main diagonal gives A332470.
Cf. A177976, A177977.

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

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)

cp's OEIS Frontend

A177517 Triangle T(n,k) read by rows defined by recurrence T(n,1)=A000007(n-1) and T(n,k) = sum_{i=1..k-1} T(n-i,k-1) if k>1.

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