Nadia Heninger - cp's OEIS frontend

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 0, 0, 1, 5, 6, 1, 0, 0, 1, 6, 10, 4, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0

Offset: 0

Nadia Heninger and N. J. A. Sloane, May 21 2010

T(n,k) = 0 if k <0 or k > n+1-k.
T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.
Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).
Row sums are A000045.
From Emanuele Munarini, May 24 2011: (Start)
Diagonal sums are A000930(n+1).
A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)
As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - Tom Copeland, Jul 02 2018

			Triangle begins:
  [1]
  [1, 1]
  [1, 2, 0]
  [1, 3, 1, 0]
  [1, 4, 3, 0, 0]
  [1, 5, 6, 1, 0, 0]
  [1, 6, 10, 4, 0, 0, 0]
  [1, 7, 15, 10, 1, 0, 0, 0]
  [1, 8, 21, 20, 5, 0, 0, 0, 0]
  [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
  [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
  [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
  [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
  [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
  [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
  [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
  [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  ...

Indranil Ghosh, Rows 0..125, flattened
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Emanuele Munarini and Norma Zagaglia Salvi, Scattered Subsets, Discrete Mathematics 267 (2003), 213-228.
Emanuele. Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Emanuele Munarini, A combinatorial interpretation of the Chebyshev polynomials, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
Peter J. Olver, The canonical contact form.
James J. Y. Zhao, Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids, arXiv:2409.08085 [math.CO], 2024. See p. 11.

Cf. A000045, A000930, A007318, A011973 (another version), A218272.
All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011
A172431 and A054142 describe the odd and even lines of the triangle.

T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Sep 16 2024 *)

create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* Emanuele Munarini, May 24 2011 */

T(n,k)=binomial(n+1-k,k) \\ Charles R Greathouse IV, Oct 24 2012

A144488 Ludwig van Beethoven, Bagatelle No. 25, "Für Elise".

Original entry on oeis.org

0, 76, 75, 76, 75, 76, 71, 74, 72, 69, 52, 57, 60, 64, 69, 71, 52, 56, 64, 68, 71, 72, 52, 57, 64, 76, 75, 76, 75, 76, 71, 74, 72, 69, 52, 57, 60, 64, 69, 71, 52, 56, 64, 72, 71, 69, 52, 57, 0, 76, 75, 76, 75, 76, 71, 74, 72, 69, 52, 57, 60, 64, 69, 71, 52, 56

Offset: 1

N. J. A. Sloane and Nadia Heninger, Jun 19 2006

Another version of A123456. The latter version sounds better when played on a MIDI player with the default settings used by the OEIS, while the present sequence is closer to the score.

N. J. A. Sloane, Table of n, a(n) for n = 1..761
Everything2.com, List of notes in MIDI format together with tempo indications [from the link below]
Everything2.com, Für Elise
N. J. A. Sloane, Listening to a sequence
Index entries for sequences related to music

Cf. A123456.

A140083 Same as A140080, except now e=6.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 2, 3, 0, 1, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

nonn

Nadia Heninger and N. J. A. Sloane, Table of n, a(n) for n = 0..5000

A140084 Same as A140080, except now e=7.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 0, 1, 2, 2, 3, 2, 3, 0, 1, 1, 2, 2, 3, 2, 0, 3, 1, 2, 3, 3, 4, 0, 1, 1, 2, 1, 2, 2, 0, 2, 3, 3, 1, 2, 3, 0, 1, 3, 4, 1, 2, 2, 0, 3, 1, 3, 1, 4, 2, 0, 1, 1, 2, 1, 2, 2, 0, 1, 2, 2, 3, 2, 3, 0, 1, 2, 3, 3, 4, 3, 0, 1, 1, 2, 3, 3, 4, 0, 1, 1, 2, 3, 4, 4, 0, 1, 1, 2, 1, 2, 3, 0

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

nonn

Nadia Heninger and N. J. A. Sloane, Table of n, a(n) for n = 0..5000

A140086 Same as A140080, except now e=9.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 1, 0, 2, 1, 2, 1, 3, 2, 1, 2, 0, 1, 2, 3, 1, 2, 2, 1, 1, 0, 3, 2, 2, 1, 1, 2, 2, 3, 0, 1, 1, 2, 2, 1, 3, 2, 1, 0, 2, 1, 2, 3, 1, 2, 1, 2, 0, 1, 3, 2, 2, 1, 2, 1, 1, 0, 1, 2, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 1, 2, 2, 3, 2, 0, 1, 1, 3, 1, 2, 2, 1, 1, 0, 1, 2, 2, 1, 2, 2, 3, 3

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

nonn

Nadia Heninger and N. J. A. Sloane, Table of n, a(n) for n = 0..5000

A140137 Numbers n such that A140080(n) = 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 23, 25, 26, 28, 29, 31, 32, 35, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 67, 70, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106, 107, 109, 110

Offset: 1

Nadia Heninger and N. J. A. Sloane, Jun 08 2008

nonn

This and A140138 together give all numbers not divisible by 3.

Cf. A140080, A140138.

A140138 Numbers n such that A140080(n) = 2.

Original entry on oeis.org

5, 10, 17, 20, 34, 40, 65, 68, 80, 130, 136, 160, 257, 260, 272, 320, 341, 514, 520, 544, 640, 682, 1025, 1028, 1040, 1088, 1109, 1280, 1301, 1349, 1361, 1364, 2050, 2056, 2080, 2176, 2218, 2560, 2602, 2698, 2722, 2728, 4097, 4100, 4112, 4160

Offset: 1

Nadia Heninger and N. J. A. Sloane, Jun 08 2008

nonn

Cf. A140080, A140137.

A140193 Same as A140191 except here e=6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 3, 3, 3, 3, 6, 6, 1, 1, 1, 1, 2, 2, 3, 3, 1, 1, 2, 2, 3, 3, 5, 5, 2, 2, 3, 3, 3, 3, 6, 6, 3, 3, 5, 5, 6, 6, 11, 11, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 3, 3, 3, 3, 6, 6, 1, 1, 3, 3, 2, 2, 5, 5, 3, 3, 6, 6, 5, 5, 11, 11, 2, 2, 3, 3, 3, 3, 6

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 06 2008

nonn

Nadia Heninger and N. J. A. Sloane, Table of n, a(n) for n = 0..3000

A140200 Partial sums of A140080.

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 5, 6, 7, 7, 9, 10, 10, 11, 12, 12, 13, 15, 15, 16, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 28, 29, 29, 30, 31, 31, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46, 47, 48, 48, 49, 51, 51, 52, 54, 54, 55, 56, 56, 57, 58

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 09 2008

nonn

A140202 Partial sums of A140082.

Original entry on oeis.org

0, 1, 2, 4, 5, 5, 7, 8, 9, 11, 11, 12, 14, 15, 16, 16, 17, 19, 21, 24, 24, 25, 26, 28, 30, 30, 31, 32, 33, 34, 34, 35, 36, 38, 40, 40, 42, 43, 46, 47, 47, 48, 49, 50, 51, 51, 53, 54, 56, 59, 59, 60, 61, 63, 64, 64, 65, 66, 67, 69, 69, 70, 71, 72, 73, 73, 75, 76, 78, 79, 79, 80, 82, 83, 84

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 09 2008

nonn

cp's OEIS Frontend

User: Nadia Heninger

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

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A144488 Ludwig van Beethoven, Bagatelle No. 25, "Für Elise".

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A140083 Same as A140080, except now e=6.

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A140084 Same as A140080, except now e=7.

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A140086 Same as A140080, except now e=9.

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A140137 Numbers n such that A140080(n) = 1.

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A140138 Numbers n such that A140080(n) = 2.

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A140193 Same as A140191 except here e=6.

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A140200 Partial sums of A140080.

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A140202 Partial sums of A140082.

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