A129183 -id:A129183 - cp's OEIS frontend

A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

1, 1, 2, 4, 8, 17, 36, 76, 162, 345, 734, 1564, 3332, 7098, 15124, 32224, 68658, 146291, 311704, 664152, 1415124, 3015237, 6424636, 13689132, 29167776, 62148513, 132421414, 282153672, 601192008, 1280975135, 2729406380, 5815615784, 12391480916, 26402844538, 56257214530, 119868682488

Offset: 0

Joerg Arndt, Jun 29 2013

nonn

What does this sequence count?
Conjectures from John Tyler Rascoe, Nov 04 2023: (Start)
a(n) is the number of integer compositions of n into two kinds of odd parts with the following restrictions. Each composition has first part 1a. For all a parts pa_i >= px_{i+1} and for all b parts pb_i >= px_{i+1} or pb_i = (p+2)a_{i+1}.
In general if B(i) = b_1, b_2, ..., b_i is an infinite sequence where b_1 > 0 and b_i <= b_{i+1} for all i, let A(q) = 1/(1-q^b_1/(1-q^b_2/(1-q^b_3/(1-...)))) be a generating function where the exponents of q are the sequence B(i).
Then A(q) counts integer compositions into parts b_i with the following restrictions. Every composition has first part p_1 = b_1 and for every pair of parts (p_j,p_{j+1}), B^-1(p_j) + 1 >= B^-1(p_{j+1}). Where j is the position of the part p_j within the composition itself and B^-1(p_j) is the index of p_j in B(i). (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A129183.

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(2*Range[nmax + 1] - 2*Floor[Range[nmax + 1]/2] - 1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
gf = 1 / G(0)
Vec(gf)

G.f.: 1/(1-q/(1-q/(1-q^3/(1-q^3/(1-q^5/(1-q^5/(1-q^7/(1-q^7/(1-...))))))))).
G.f.: 1/W(0), where W(k)= 1 - x^(2*k+1)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - Vaclav Kotesovec, Sep 05 2017
Conjecture: a(n) = Sum_{i=0..floor((n-sqrt(2*n-1))/2)} A129183(n-(2*i),n-i). - John Tyler Rascoe, Nov 04 2023

A140717 Triangle read by rows: T(n,k) is the number of Dyck paths d of semilength n such that sum of peakheights of d - number of peaks of d = k (n >= 0, 0 <= k <= floor(n^2/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 1, 4, 9, 12, 10, 4, 2, 1, 5, 14, 25, 31, 26, 16, 9, 4, 1, 1, 6, 20, 44, 70, 82, 74, 54, 38, 22, 12, 4, 2, 1, 7, 27, 70, 134, 196, 227, 215, 179, 139, 99, 64, 38, 20, 9, 4, 1, 1, 8, 35, 104, 231, 400, 558, 644, 641, 576, 488, 384, 288, 200, 134, 80

Offset: 0

Emeric Deutsch, Jun 08 2008

T(n,k) is the number of 321-avoiding permutations of {1,2,...,n} having inversion number equal to k. Example: T(4,2) = 5 because we have 1423, 1342, 3124, 2143 and 2341.

Conjecture: antidiagonal sums equal A227309. - Mikhail Kurkov, Aug 30 2024

			T(4,2) = 5 because we have UDUUDUDD (5 - 3 = 2), UDUUUDD (4 - 2 = 2), UUDDUUDD (4 - 2 = 2), UUDUDDUD (5 - 3 = 2) and UUUDDDUD (4 - 2 = 2); here U = (1,1), D = (1,-1).
Triangle starts:
  1;
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,  4,  1;
  1, 4,  9, 12, 10,  4,  2;
  1, 5, 14, 25, 31, 26, 16, 9, 4, 1;

Alois P. Heinz, Rows n = 0..50, flattened
E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, ECO: a methodology for the enumeration of combinatorial objects, Journal of Difference Equations and Applications, 5, 1999, 435-490.
E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, Some permutations with forbidden subsequences and their inversion number, Discrete Math., 234, 2001, 1-15.
Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202 (see section 5).
G. Feinberg and K.-H. Lee, Homogeneous representations of KLR-algebras and fully commutative elements, arXiv preprint arXiv:1401.0845 [math.RT], 2014-2015.
Niket Gowravaram and Tanya Khovanova, On the Structure of nil-Temperley-Lieb Algebras of type A, arXiv:1509.00462 [math.CO], 2015.

Row sums are the Catalan numbers A000108.

Cf. A008549, A129183, A227309, A227543.

H := 1/(1+z-t*x*z-z*h[1]):
for n to 13 do h[n]:=1/(1+z-x*t^(n+1)*z-z*h[n+1]) end do:
G := subs({h[11]=0,x=1/t},H): Gser := simplify(series(G,z=0,12)):
for n from 0 to 9 do P[n] := sort(coeff(Gser,z,n)) end do:
for n from 0 to 9 do seq(coeff(P[n],t,j), j=0..floor((1/4)*n^2)) end do;
# yields sequence in triangular form

m = rows = 10; mt = 2 m + 1; mx = mz = m - 1;
H[, , ] = 0; Do[H[t, x_, z_] = Series[1 + z (H[t, t x, z] - 1 + t x) H[t, x, z], {t, 0, mt}, {x, 0, mx}, {z, 0, mz}] // Normal, {m}];
G[t_, z_] = Series[H[t, 1/t, z], {t, 0, mt}, {z, 0, mz}] // Normal // Collect[#, z]&;
CoefficientList[#, t]& /@ CoefficientList[G[t, z], z] // Take[#, m]& // Flatten (* Jean-François Alcover, Nov 25 2018 *)

G.f.: G(t,z) = H(t,1/t,z), where H(t,x,z) = 1 + zH(t,x,z)[H(t,tx,z)-1+tx] (H(t,x,z) is the trivariate g.f. of Dyck paths with respect to semilength, sum of peak-heights and number of peaks, marked by z, t and x, respectively).
Sum_{k>=0} k*T(n,k) = A008549(n-1).
Row n has 1 + floor(n^2/4) entries.
Conjecture: n-th row polynomial equals t_n for n > 0 where we start with vector v of fixed length m with elements v_i = 1, then set t = v and for i=1..m-1, for j=i+1..m apply [v_i, v_j] := [v_i + z^(j-i)*v_j, z*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order) and t_{i+1} := v_{i+1} (after ending each cycle for j). It also looks like that if we change z^(j-i) to z^(2*(j-i)) it gives us equivalence of t_n and n-th row polynomial of A227543. - Mikhail Kurkov, Aug 30 2024

A201161 Irregular triangle read by rows: number of colored Motzkin paths of length n and rank k.

Original entry on oeis.org

1, 2, 1, 4, 2, 4, 8, 1, 4, 9, 12, 16, 2, 4, 12, 20, 30, 32, 32, 1, 4, 9, 20, 34, 56, 73, 88, 80, 64, 2, 4, 12, 24, 46, 72, 116, 156, 206, 232, 240, 192, 128

Offset: 0

N. J. A. Sloane, Nov 27 2011

Perhaps the row-reversed variant of A129183. - R. J. Mathar, Dec 13 2011

			Triangle begins
1
2
1 4
2 4 8
1 4 9 12 16
2 4 12 20 30 32 32
1 4 9 20 34 56 73 88 80 64
2 4 12 24 46 72 116 156 206 232 240 192 128
...

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See Section 4.

cp's OEIS Frontend

A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

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A140717 Triangle read by rows: T(n,k) is the number of Dyck paths d of semilength n such that sum of peakheights of d - number of peaks of d = k (n >= 0, 0 <= k <= floor(n^2/4)).

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A201161 Irregular triangle read by rows: number of colored Motzkin paths of length n and rank k.

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