A077419 -id:A077419 - cp's OEIS frontend

A200454 T(n,k)=Number of -k..k arrays x(0..n+1) of n+2 elements with zero sum and nonzero first and second differences.

4, 10, 4, 26, 44, 6, 44, 142, 144, 10, 72, 342, 728, 486, 14, 102, 678, 2334, 3788, 1582, 22, 142, 1148, 5720, 16380, 19802, 5478, 34, 184, 1832, 12002, 50852, 115140, 103726, 18692, 52, 236, 2744, 22276, 127988, 451708, 820650, 548204, 64782, 80, 290, 3874

Offset: 1

R. H. Hardin Nov 18 2011

Table starts
...4.....10.......26.........44..........72..........102...........142
...4.....44......142........342.........678.........1148..........1832
...6....144......728.......2334........5720........12002.........22276
..10....486.....3788......16380.......50852.......127988........278906
..14...1582....19802.....115140......451708......1375006.......3513884
..22...5478...103726.....820650.....4062384.....14923636......44716536
..34..18692...548204....5876818....36725772....163058296.....572857272
..52..64782..2916664...42324384...334032710...1791564880....7380022092
..80.223272.15576706..306098316..3050654456..19771609900...95496980144
.126.776430.83481240.2222013090.27965763262.219008350922.1240352594210

			Some solutions for n=4 k=3
..0....1...-1....1....1...-1....0....2...-3...-1...-1....0...-1...-1....3...-3
.-1...-3....2....3...-1....1...-1....0....3...-2....1...-2....3....3...-3....2
..3...-1...-1...-3....2...-1...-3...-1....1....0...-3....0....2....0...-2...-1
.-1....3....1...-1...-2....1....2...-3....2....1....0...-3...-2...-1....1....3
..1...-2...-3....2....1....2....0....0...-3....3....2....3....1...-3....0....0
.-2....2....2...-2...-1...-2....2....2....0...-1....1....2...-3....2....1...-1

R. H. Hardin, Table of n, a(n) for n = 1..715

Column 1 is twice A077419(n+2)

A200057 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 10, 4, 1, 8, 22, 26, 6, 1, 10, 36, 78, 68, 10, 1, 12, 56, 172, 288, 178, 14, 1, 14, 78, 324, 840, 1098, 472, 22, 1, 16, 106, 546, 1948, 4172, 4224, 1276, 34, 1, 18, 136, 850, 3914, 11962, 20978, 16432, 3462, 52, 1, 20, 172, 1252, 7074, 28554, 74338

Offset: 1

R. H. Hardin Nov 13 2011

Table starts
..1....1......1.......1........1........1.........1..........1..........1
..2....4......6.......8.......10.......12........14.........16.........18
..4...10.....22......36.......56.......78.......106........136........172
..4...26.....78.....172......324......546.......850.......1252.......1764
..6...68....288.....840.....1948.....3914......7074......11862......18732
.10..178...1098....4172....11962....28554.....59910.....114232.....202314
.14..472...4224...20978....74338...211242....514168....1115572....2215290
.22.1276..16432..106674...466548..1577878...4453946...10995240...24477966
.34.3462..64310..545698..2947742.11867186..38855488..109147062..272432422
.52.9496.253692.2811236.18746754.89815404.341052122.1090022120.3050199016

			Some solutions for n=7 k=6
..3....0...-2....1....0...-3....0....0...-3....0...-3...-3....3...-3....0...-6
.-1...-3...-6....0...-1....6....2....5...-6....5...-6....0...-3...-2...-4....3
..0....5....6....3....6...-3...-2...-5....4...-5....3...-1....3...-3....4...-2
.-4....2...-5...-4...-3...-2....1....6...-5....6...-3....4...-5....1...-6....5
..4....4....1...-1...-1...-4...-5...-3....5...-3....6...-1....4...-1....6....1
.-3...-6....0...-3...-6....6....4....1...-1....2...-3....3...-3....6...-5....3
..1...-2....6....4....5....0....0...-4....6...-5....6...-2....1....2....5...-4

R. H. Hardin, Table of n, a(n) for n = 1..871

Column 1 is twice A077419 for n>1

Row 3 is twice A171769

A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 1

Offset: 0

N. J. A. Sloane, Jan 19 2003

Row sums are Fibonacci numbers A000045. - Roger L. Bagula, Oct 07 2006

This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008

			Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{1, 2, 1, 1},
{1, 2, 2, 2, 1},
{1, 3, 3, 3, 2, 1},
{1, 3, 4, 5, 4, 3, 1},
{1, 4, 6, 7, 7, 5, 3, 1},
{1, 4, 7, 10, 11, 10, 7, 4, 1},
{1, 5, 10, 14, 17, 16, 13, 8, 4, 1},
{1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}

Giovanni Resta, Rows n=0..139 of triangle, flattened
Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020.
A. Khrabrov and K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170 [math.CO], 2018-2020.
Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, (2019) [math].
Sophie Morier-Genoud and Valentin Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes, arXiv:2011.10809 [math.QA], 2020.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, arXiv:2503.23834 [math.CO], 2025. See p. 7.
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.
Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.

Largest element in each row gives A077419.

p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = Expand@ If[Mod[k, 2] == 1, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]]; Flatten[ Table[CoefficientList[p[n, x], x], {n, 0, 10}]] (* Roger L. Bagula, Oct 07 2006 *)
T[ n_, k_] := (T[n, k] = Which[k<0 || k>n, 0, k==0, 1, True, T[n-1, k-Mod[n, 2]] + T[n-2, k-Mod[n+1, 2]*2]]); (* Michael Somos, Dec 12 2023 *)

{T(n, k) = if(k<0 || k>n, 0, k==0, 1, T(n-1, k-(n%2)) + T(n-2, k-(n+1)%2*2))}; /* Michael Somos, Dec 12 2023 */

Define polynomials by: if k is odd then p(k, x) = x*p(k - 1, x) + p(k - 2, x); if k is even then: p(k, x) = p(k - 1, x) + x^2*p(k - 2, x). Triangle gives array of coefficients. - Roger L. Bagula, Oct 07 2006

Mma program editing and a(67)-a(79) from Giovanni Resta, May 26 2015

A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 0, 0, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 0, 0, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 0, 0, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 0, 0

Offset: 1

Clark Kimberling, Dec 07 2002

Row sums: 1,1,2,3,5,8,13,..., the Fibonacci numbers (A000045).

			T(6,2) counts the words 010001 and 000101. Top of triangle:
1 = T(1,0)
0 1 = T(2,0) T(2,1)
1 1 0
0 1 1 1
1 2 1 1 0

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Cf. A078808, A078821, A079487, A123245, A077419.

T(n, k)=T(n-1, n-k-1)+T(n-3, n-k-3)+...+T(n-2m-1, n-k-2m-1), where m=[(n-1)/2] and (by definition) T(i, j)=0 if i<0 or j<0 or i=j.

Row 0 removed to stick to the triangle format by Andrey Zabolotskiy, Sep 22 2017

A123245 Triangle A079487 with reversed rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 1, 1, 3, 4, 5, 4, 3, 1, 1, 3, 5, 7, 7, 6, 4, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 4, 8, 13, 16, 17, 14, 10, 5, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1

Offset: 0

Roger L. Bagula, Oct 07 2006

Row sums give Fibonacci numbers (A000045).

			{1},
{1, 1},
{1, 1, 1},
{1, 1, 2, 1},
{1, 2, 2, 2, 1},
{1, 2, 3, 3, 3, 1},
{1, 3, 4, 5, 4, 3, 1},
{1, 3, 5, 7, 7, 6, 4, 1},
{1, 4, 7, 10, 11, 10, 7, 4, 1},
{1, 4, 8, 13, 16, 17, 14, 10, 5, 1},
{1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}

Sébastien Labbé and Mélodie Lapointe, The q-analog of the Markoff injectivity conjecture over the language of a balanced sequence, Comb. Theor. (2022) Vol. 2, No. 1, #9. See also arXiv:2106.15886 [math.CO], 2021.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170 [math.CO], 2018-2020.
Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, (2019) [math].
Sophie Morier-Genoud and Valentin Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes, arXiv:2011.10809 [math.QA], 2020.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, arXiv:2503.23834 [math.CO], 2025. See p. 7.
Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.

Cf. A000045, A079487, A078807, A078808, A077419.

p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = If[Mod[k, 2] == 0, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]];
Table[CoefficientList[p[n, x], x], {n, 0, 10}] // Flatten

p(k, x) = x*p(k - 1, x) + p(k - 2, x) for k even, otherwise p(k, x) = p(k - 1, x) + x^2*p(k - 2, x).

Edited by Joerg Arndt, May 26 2015

Offset corrected by Andrey Zabolotskiy, Sep 22 2017

cp's OEIS Frontend

A200454 T(n,k)=Number of -k..k arrays x(0..n+1) of n+2 elements with zero sum and nonzero first and second differences.

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A200057 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and elements alternately strictly increasing and strictly decreasing.

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A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.

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A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

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A123245 Triangle A079487 with reversed rows.

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