1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256, 0, 10, 0, -160, 0, 672, 0, -1024, 0, 512, -1, 0, 60, 0, -560, 0, 1792, 0, -2304, 0, 1024, 0, -12, 0, 280, 0, -1792, 0, 4608, 0, -5120, 0, 2048, 1, 0, -84, 0, 1120, 0, -5376, 0, 11520, 0, -11264, 0, 4096
Offset: 0
Triangle begins:
1;
0, 2;
-1, 0, 4;
0, -4, 0, 8;
1, 0, -12, 0, 16;
...
E.g., fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x) = -4*x + 8*x^3.
A091894
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k ddu's [here u = (1,1) and d = (1,-1)].
Original entry on oeis.org
1, 1, 2, 4, 1, 8, 6, 16, 24, 2, 32, 80, 20, 64, 240, 120, 5, 128, 672, 560, 70, 256, 1792, 2240, 560, 14, 512, 4608, 8064, 3360, 252, 1024, 11520, 26880, 16800, 2520, 42, 2048, 28160, 84480, 73920, 18480, 924, 4096, 67584, 253440, 295680, 110880, 11088, 132
Offset: 0
T(4,1) = 6 because we have uduu(ddu)d, uu(ddu)dud, uuu(ddu)dd, uu(ddu)udd, uudu(ddu)d and uuud(ddu)d [here u = (1,1), d = (1,-1) and the ddu's are shown between parentheses].
Triangle begins:
1;
1;
2;
4, 1;
8, 6;
16, 24, 2;
32, 80, 20;
64, 240, 120, 5;
128, 672, 560, 70;
256, 1792, 2240, 560, 14;
...
- T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 4.3.
- Alois P. Heinz, Rows n = 0..200, flattened
- Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 7.
- Jean-Luc Baril, Sergey Kirgizov, and Vincent Vanjovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, Disc. Math. 341 (2018) 2608-2615, Table 1.
- Andrew M. Baxter, Refining enumeration schemes to count according to permutation statistics, arXiv:1401.0337 [math.CO], 2014.
- Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
- David Callan, A variant of Touchard's Catalan number identity, arXiv:1204.5704 [math.CO], 2012. - _N. J. A. Sloane_, Oct 10 2012
- David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
- Colin Defant, Postorder Preimages, arXiv:1604.01723 [math.CO], 2016.
- Colin Defant, Stack-Sorting Preimages of Permutation Classes, arXiv:1809.03123 [math.CO], 2018.
- Colin Defant, Counting 3-Stack-Sortable Permutations, arXiv:1903.09138 [math.CO], 2019.
- Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 22.
- Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202 (see pp. 192-193)..
- Maciej Dziemiańczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
- Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
- K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.
- Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018. [Broken link]
- John Riordan, A Note on Catalan Parentheses, Amer. Math. Monthly, Vol. 80, No. 8 (1973), pp. 904-906.
- Tom Roberts and Thomas Prellberg, Improving Convergence of Generalised Rosenbluth Sampling for Branched Polymer Models by Uniform Sampling, arXiv:2401.12201 [cond-mat.stat-mech], 2024. See p. 12.
- A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
- L. W. Shapiro, A short proof of an identity of Touchard's concerning Catalan numbers, J. Combin. Theory Ser. A 20 (1976) 375-376.
- Guoce Xin and Jing-Feng Xu, A short approach to Catalan numbers modulo 2^r, Electronic Journal of Combinatorics Vol. 18 Issue 1 (2011), #P177 (see page 2).
- Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
- Index entries for sequences related to Łukasiewicz.
-
T:=Concatenation([1],Flat(List([1..13],n->List([0..Int((n-1)/2)],k->2^(n-2*k-1)*Binomial(n-1,2*k)*Binomial(2*k,k)/(k+1))))); # Muniru A Asiru, Nov 29 2018
-
a := proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)*binomial(n-1,2*k)*binomial(2*k,k)/(k+1) fi end: 1,seq(seq(a(n,k),k=0..(n-1)/2),n=1..15);
-
A091894[n_] := Prepend[Table[ CoefficientList[ 2^i (1 - z)^((2 i + 3)/2) Hypergeometric2F1[(i + 3)/2, (i + 4)/2, 2, z], z], {i, 0, n}], {1}] (* computes a table of the first n rows. Stumbled accidentally on it. Perhaps someone can find a relationship here? Thies Heidecke (theidecke(AT)astrophysik.uni-kiel.de), Sep 23 2008 *)
Join[{1},Select[Flatten[Table[2^(n-2k-1) Binomial[n-1,2k] Binomial[2k,k]/ (k+1), {n,20},{k,0,n}]],#!=0&]] (* Harvey P. Dale, Mar 05 2012 *)
p[n_] := 2^n Hypergeometric2F1[(1 - n)/2, -n/2, 2, x]; Flatten[Join[{{1}}, Table[CoefficientList[p[n], x], {n, 0, 12}]]] (* Peter Luschny, Jan 23 2018 *)
-
{T(n, k) = if( n<1, n==0 && k==0, polcoeff( polcoeff( serreverse( x / (1 + 2*x*y + x^2) + x * O(x^n)), n), n-1 - 2*k))} /* Michael Somos, Sep 25 2006 */
-
[1] + [[2^(n-2*k-1)*binomial(n-1,2*k)*binomial(2*k,k)/(k+1) for k in (0..floor((n-1)/2))] for n in (1..12)] # G. C. Greubel, Nov 30 2018
A128099
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
Original entry on oeis.org
1, 1, 1, 2, 1, 4, 1, 6, 4, 1, 8, 12, 1, 10, 24, 8, 1, 12, 40, 32, 1, 14, 60, 80, 16, 1, 16, 84, 160, 80, 1, 18, 112, 280, 240, 32, 1, 20, 144, 448, 560, 192, 1, 22, 180, 672, 1120, 672, 64, 1, 24, 220, 960, 2016, 1792, 448, 1, 26, 264, 1320, 3360, 4032, 1792, 128, 1, 28
Offset: 0
Triangle starts:
1;
1;
1, 2;
1, 4;
1, 6, 4;
1, 8, 12;
1, 10, 24, 8;
1, 12, 40, 32;
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358
- G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
- Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
- Richard Fors, Independence Complexes of Certain Families of Graphs, Master's thesis in Mathematics at KTH, presented Aug 19 2011.
- R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 [math.CO] (2016).
- Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
- Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n
- Eric Weisstein's World of Mathematics, Jacobsthal Polynomial
-
T := proc(n,k) if k<=n/2 then 2^k*binomial(n-k,k) else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
T := proc(n, k) option remember: if k<0 or k > floor(n/2) then return(0) fi: if k = 0 then return(1) fi: 2*procname(n-2, k-1) + procname(n-1, k): end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..13); # Johannes W. Meijer, Aug 28 2013
-
Table[2^k*Binomial[n - k, k] , {n,0,25}, {k,0,Floor[n/2]}] // Flatten (* G. C. Greubel, Dec 28 2016 *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
A133156
Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.
Original entry on oeis.org
1, 2, 4, -1, 8, -4, 16, -12, 1, 32, -32, 6, 64, -80, 24, -1, 128, -192, 80, -8, 256, -448, 240, -40, 1, 512, -1024, 672, -160, 10, 1024, -2304, 1792, -560, 60, -1, 2048, -5120, 4608, -1792, 280, -12, 4096, -11264, 11520, -5376, 1120, -84, 1
Offset: 0
The first few Chebyshev polynomials of the second kind are
1;
2x;
4x^2 - 1;
8x^3 - 4x;
16x^4 - 12x^2 + 1;
32x^5 - 32x^3 + 6x;
64x^6 - 80x^4 + 24x^2 - 1;
128x^7 - 192x^5 + 80x^3 - 8x;
256x^8 - 448x^6 + 240x^4 - 40x^2 + 1;
512x^9 - 1024x^7 + 672x^5 - 160x^3 + 10x;
...
From _Roger L. Bagula_ and _Gary W. Adamson_: (Start)
1;
2;
4, -1;
8, -4;
16, -12, 1;
32, -32, 6;
64, -80, 24, -1;
128, -192, 80, -8;
256, -448, 240, -40, 1;
512, -1024, 672, -160, 10;
1024, -2304, 1792, -560, 60, -1; (End)
From _Philippe Deléham_, Dec 27 2011: (Start)
Triangle (2, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, ...) begins:
1;
2, 0;
4, -1, 0;
8, -4, 0, 0;
16, -12, 1, 0, 0;
32, -32, 6, 0, 0, 0;
64, -80, 24, -1, 0, 0, 0; (End)
- Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian Transformation, Journal of Integer Sequences, Vol. 12 (2009), page 25
- P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
- Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120-135. MR3149030
- Caglar Koca and Ozgur B. Akan, Modelling 1D Partially Absorbing Boundaries for Brownian Molecular Communication Channels, arXiv:2402.15888 [q-bio.MN], 2024. See p. 9.
- Wikipedia, Chebyshev polynomials
-
t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Dec 19 2008 *)
A173284
Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1
Offset: 0
First few rows of the triangle:
1;
1;
2, 1;
3, 1;
5, 2, 1;
8, 3, 1;
13, 5, 2, 1;
21, 8, 3, 1;
34, 13, 5, 2, 1;
55, 21, 8, 3, 1;
89, 34, 13, 5, 2, 1;
144, 55, 21, 8, 3, 1;
233, 89, 34, 13, 5, 2, 1;
377, 144, 55, 21, 8, 3, 1;
610, 233, 89, 34, 13, 5, 2, 1;
...
-
T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013
A209634
Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.
Original entry on oeis.org
1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10
Offset: 1
Triangle:
1
4
7, 1
10, 4
13, 7, 1
16, 10, 4
19, 13, 7, 1
22, 16, 10, 4
25, 19, 13, 7, 1
28, 22, 16, 10, 4
...
-
T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013
A317494
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 4, 1, 10, 12, 1, 12, 24, 1, 14, 40, 8, 1, 16, 60, 32, 1, 18, 84, 80, 1, 20, 112, 160, 16, 1, 22, 144, 280, 80, 1, 24, 180, 448, 240, 1, 26, 220, 672, 560, 32, 1, 28, 264, 960, 1120, 192, 1, 30, 312, 1320, 2016, 672, 1, 32, 364, 1760, 3360, 1792, 64
Offset: 0
Triangle begins:
1;
1;
1;
1, 2;
1, 4;
1, 6;
1, 8, 4;
1, 10, 12;
1, 12, 24;
1, 14, 40, 8;
1, 16, 60, 32;
1, 18, 84, 80;
1, 20, 112, 160, 16;
1, 22, 144, 280, 80;
1, 24, 180, 448, 240;
1, 26, 220, 672, 560, 32;
1, 28, 264, 960, 1120, 192;
1, 30, 312, 1320, 2016, 672;
1, 32, 364, 1760, 3360, 1792, 64;
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359
-
Flat(List([0..20],n->List([0..Int(n/3)],k->2^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018
-
t[n_, k_] := t[n, k] = 2^k/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten
A317495
Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
Original entry on oeis.org
1, 2, 4, 8, 1, 16, 4, 32, 12, 64, 32, 1, 128, 80, 6, 256, 192, 24, 512, 448, 80, 1, 1024, 1024, 240, 8, 2048, 2304, 672, 40, 4096, 5120, 1792, 160, 1, 8192, 11264, 4608, 560, 10, 16384, 24576, 11520, 1792, 60, 32768, 53248, 28160, 5376, 280, 1, 65536, 114688, 67584, 15360, 1120, 12
Offset: 0
Triangle begins:
1;
2;
4;
8, 1;
16, 4;
32, 12;
64, 32, 1;
128, 80, 6;
256, 192, 24;
512, 448, 80, 1;
1024, 1024, 240, 8;
2048, 2304, 672, 40;
4096, 5120, 1792, 160, 1;
8192, 11264, 4608, 560, 10;
16384, 24576, 11520, 1792, 60;
32768, 53248, 28160, 5376, 280, 1;
65536, 114688, 67584, 15360, 1120, 12;
131072, 245760, 159744, 42240, 4032, 84;
262144, 524288, 372736, 112640, 13440, 448, 1;
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.
-
Flat(List([0..20],n->List([0..Int(n/3)],k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018
-
/* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018
-
t[n_, k_] := t[n, k] = 2^(n - 3k)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten
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