A227543 -id:A227543 - cp's OEIS frontend

A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 2, 2, 2, 1, 1, 0, -1, -2, -2, -4, -4, -5, -5, -6, -5, -6, -4, -4, -3, -2, 1, 1, 4, 5, 8, 9, 12, 12, 15, 15, 17, 16, 18, 15, 16, 13, 13, 8, 7, 1, 0, -7, -9, -17, -19, -27, -29, -37, -38, -46, -46, -53, -51, -57, -53, -57, -51, -53, -45, -45, -32, -31

Offset: 0

Paul D. Hanna, Jul 24 2013

Conjecture: a(n+1) = A286744(n) - A286745(n). - George Beck May 13 2017

			G.f.: A(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...
where
A(x) = 1 - x^2/(1-x) + x^6/((1-x)*(1-x^2)) - x^12/((1-x)*(1-x^2)*(1-x^3)) + x^20/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) - x^30/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Paul D. Hanna)

Cf. A227620, A227543, A039924, A005169, A286744, A286745, A003106.

a(n)=polcoeff(sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k,1+x*O(x^n))),n)
for(n=0, 80, print1(a(n), ", "))

A326453 Triangle read by rows: T(n,k) is the number of small Schröder paths of semilength k such that the area between the path and the x-axis is equal to n (n >= 0; 0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 3, 1, 0, 0, 0, 2, 6, 4, 1, 0, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 1, 6, 16, 15, 6, 1, 0, 0, 0, 1, 5, 19, 30, 21, 7, 1, 0, 0, 0, 0, 5, 19, 45, 50, 28, 8, 1, 0, 0, 0, 0, 4, 19, 55, 90, 77, 36, 9, 1, 3, 19, 61, 131, 161, 112, 45, 10, 1

Offset: 0

Peter Bala, Jul 06 2019

A239927 is the companion triangle for Dyck paths.
A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.
The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles are of two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). A small Schröder path of semilength k has k up-triangles in the bottom row of its stack. See the illustration in the Links section for an example. Thus an alternative description of the triangle entry T(n,k) is the number of n triangle stacks, in the sense of A224704, containing k up-triangles in the bottom row.

			Triangle begins
  n\k|  0    1   2    3    4    5    6    7   8    9
  --------------------------------------------------
   0 |  1
   1 |  0    1
   2 |  0    0   1
   3 |  0    0   1    1
   4 |  0    0   1    2    1
   5 |  0    0   0    3    3    1
   6 |  0    0   0    2    6    4    1
   7 |  0    0   0    1    7   10    5    1
   8 |  0    0   0    1    6   16   15    6   1
   9 |  0    0   0    1    5   19   30   21   7   1
   ...
Example of a stack of 10 up- and down-triangles with 5 up-triangles in the bottom row.
          /\  /\
         /__\/__\     __
        /\  /\  /\  /\  /\
       /__\/__\/__\/__\/__\

P. Bala, Illustration for row 5
P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454

Row sums A224704. Cf. A047998, A227543, A239927, A309086, A326454.

O.g.f. as a continued fraction: A(q,u) = 1/(1 + u - (1 + q)*u/(1 + u - (1 + q^3)*u/(1 + u - (1 + q^5)*u/( (...) )))) =  1 + q*u + q^2*u^2 + q^3*(u^2 + u^3) + q^4*(u^2 + 2*u^3 + u^4) + ...(q marks the area, u marks the up- triangles in the bottom row).
Alternative forms: A(q,u) = 1/(1 - q*u/(1 - q^2*u - q^3*u/(1 - q^4*u/( (...) ))));
A(q,u) = 1/(1 - q*u/(1 - (q^2 + q^3)*u/(1 - q^5*u/(1 - (q^4 + q^7)*u/(1 - q^9*u/(1 - (q^6 + q^11)*u/(1 - q^13*u/( (...) )))))))).
O.g.f. as a ratio of q-series: N(q,u)/D(q,u), where N(q,u) = Sum_{n >= 0} (-1)^n*u^n*q^(2*n^2 + n)/( (1 - q^2)*(1 - q^4)*...*(1 - q^(2*n)) * (1 - u*q^2)*(1 - u*q^4)*...*(1 - u*q^(2*n)) ) and D(q,u) = Sum_{n >= 0} (-1)^n*u^n*q^(2*n^2 - n)/( (1 - q^2)*(1 - q^4)*...*(1 - q^(2*n)) * (1 - u*q^2)*(1 - u*q^4)*...*(1 - u*q^(2*n)) ).

A326454 Irregular triangle read by rows: T(n,k) is the number of small Schröder paths such that the area between the path and the x-axis is equal to n and contains k down-triangles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 7, 5, 1, 9, 13, 1, 1, 11, 25, 8, 1, 13, 41, 28, 1, 1, 15, 61, 68, 11, 1, 17, 85, 136, 51, 1, 1, 19, 113, 240, 155, 15, 1, 21, 145, 388, 371, 86, 1, 1, 23, 181, 588, 763, 314, 19

Offset: 0

Peter Bala, Jul 06 2019

A227543 is the companion triangle for Dyck paths.
Number of n triangle stacks, in the sense of A224704, containing k down- triangles.
A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.
The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). See the illustration in the Links section for an example.

			Triangle begins
  n\k|  0    1    2     3    4
------------------------------
   0 |  1
   1 |  1
   2 |  1
   3 |  1    1
   4 |  1    3
   5 |  1    5    1
   6 |  1    7    5
   7 |  1    9   13    1
   8 |  1   11   25    8
   9 |  1   13   41   28    1
  10 |  1   15   61   68   11
  ...

P. Bala, Illustration for row 5
P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454

Row sums A224704. Cf. A001003, A227543, A309086, A326453.

O.g.f. as a continued fraction: A(q,d) = 1/(2 - (1 + q)/(2 - (1 + q^3*d)/(2 - (1 + q^5*d^2)/( (...) )))) =  1 + q + q^2 + q^3*(1 + d) + q^4*(1 + 3*d) + q^5*(1 + 5*d + d^2) + ... (q marks the area, d marks down-triangles).
Other continued fractions: A(q,d) = 1/(1 - q/(1 - q^2*d - q^3*d/(1 - q^4*d^2 - q^5*d^2/(1 - q^6*d^3 - (...) )))).
A(q,d) = 1/(1 - q/(1 - (q^2*d + q^3*d)/(1 - q^5*d^2/(1 - (q^4*d^2 + q^7*d^3)/(1 - q^9*d^4/(1 - (q^6*d^3 + q^11*d^5)/(1 - q^13*d^6/( (...) )))))))).
O.g.f. as a ratio of q-series: N(q,d)/D(q,d), where N(q,d) = Sum_{n >= 0} (-1)^n*d^(n^2)*q^(2*n^2 + n)/( (1 - d*q^2)*(1 - d^2*q^4)*...*(1 - d^n*q^(2*n)) )^2 and D(q,d) = Sum_{n >= 0} (-1)^n*d^(n^2 - n)*q^(2*n^2 - n)/( (1 - d*q^2)*(1 - d^2*q^4)*...*(1 - d^n*q^(2*n)) )^2.

A274291 The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 17, 44, 118, 338, 1003, 3039, 9466, 30009, 96757, 316429, 1047683, 3511473, 11876457, 40537388, 139490014, 483393651, 1686007017, 5917253784, 20879801881, 74038098051, 263793988890, 943928231920, 3390975927021, 12227214763162, 44242758258306

Offset: 0

N. J. A. Sloane, Jun 17 2016

nonn

Previous name was: The width of the lattice E_n defined in the paper by Boldi and Vigna, that is, the cardinality of a maximal antichain.

a(n) is the maximum entry in row n of the triangle T(n,k) defined in A138158, or equivalently, the maximum entry in row n of the triangle T(n,k) defined in A227543. All level sizes of the lattice are given by A138158 and A227543. - Torsten Muetze, Nov 28 2018

			For n=4 there are 14 Dyck paths, and 1,3,3,3,2,1,1 of them have area 0,1,2,3,4,5,6, respectively, where the area is normalized to the range 0,...,n(n-1)/2. These Dyck paths are UDUDUDUD (area=0), UUDDUDUD, UDUUDDUD, UDUDUUDD (area=1), UUDUDDUD, UDUUDUDD, UUDDUUDD (area=2), UUUDDDUD, UUDUDUDD, UDUUUDDD (area=3), UUUDDUDD, UUDUUDDD (area=4), UUUDUDDD (area=5), UUUUDDDD (area=6). The maximum among the numbers 1,3,3,3,2,1,1 is 3, so a(4)=3.

Winston, Kenneth J., and Daniel J. Kleitman. "On the asymptotic number of tournament score sequences." Journal of Combinatorial Theory, Series A 35.2 (1983): 208-230. See Table 1.

Torsten Muetze, Table of n, a(n) for n = 0..300
Paolo Boldi and Sebastiano Vigna, On the Lattice of Antichains of Finite Intervals, Order (2016), 1-25.
Paolo Boldi, Sebastiano Vigna, On the lattice of antichains of finite intervals, arXiv preprint arXiv:1510.03675 [math.CO], 2015-2016.

Cf. A138158, A227543.

a(0)=1 inserted by Sebastiano Vigna, Dec 20 2017

New name and more terms from Torsten Muetze, Nov 28 2018

A309086 Irregular triangle read by rows: T(n,k) is the number of small Schröder paths of semilength n such that the area between the path and the x-axis contains k down-triangles.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 4, 2, 1, 6, 12, 12, 8, 4, 2, 1, 8, 24, 38, 40, 32, 24, 16, 8, 4, 2, 1, 10, 40, 88, 128, 140, 130, 112, 88, 64, 44, 28, 16, 8, 4, 2, 1, 12, 60, 170, 3320, 448, 512, 520, 488, 428, 358, 288, 220, 160, 112, 76, 48, 28, 16, 8, 4, 2

Offset: 0

Peter Bala, Jul 16 2019

A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.

The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). These are the triangle stacks of A224704. Here we enumerate triangle stacks with n >= 1 up-triangles in the bottom row of the stack (corresponding to small Schröder paths of semilength n) and containing k >= 0 down-triangles in the stack. See the illustration in the Links section for an example.

			   n\k |  0    1    2    3    4    5    6    7   8   9  10
   - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    0  |  1
    1  |  1
    2  |  1    2
    3  |  1    4    4    2
    4  |  1    6   12   12    8    4    2
    5  |  1    8   24   38   40   32   24   16   8   4   2
   ...

P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454
P. Bala, Illustration for row 3

Row sums A001003. Cf. A224704, A227543, A326453, A326454.

O.g.f. as a continued fraction: A(u,d) = 1/(1 - u/(1 - u*d - u*d/(1 - u*d^2 - u*d^2/(1 - u*d^3 -  (...) )))) =  1 + u + (1 + 2*d)*u^2 + (1 + 4*d + 4*d^2 + 2*d^3)*u^3 + ... (u marks the semilength of the path (or, equivalently, up-triangles in the bottom row of the associated triangle stack) and d marks down-triangles in the stack).
Other continued fractions: A(u,d) = 1/(1 + u - 2*u/(1 + u - (1 + d)*u/(1 + u - (1 + d^2)*u/(1 + u - (...) )))).
A(u,d) = 1/(1 - u/(1 - (d + d)*u/(1 - d^2*u/(1 - (d^2 + d^3)*u/(1 - d^4*u/(1 - (d^3 + d^5)*u/(1 - d^6*u/(1 - (d^4 + d^7)*u/(1 - (...) ))))))))).
O.g.f. as a ratio of q-series: N(u,d)/D(u,d), where N(u,d) = Sum_{n >= 0} (-1)^n*u^n*d^(n^2)/( (1 - d)*(1 - d^2)*...*(1 - d^n) * (1 - u*d)*(1 - u*d^2)*...*(1 - u*d^n) ) and D(u,d) =  Sum_{n >= 0} (-1)^n*u^n*d^(n(n-1))/( (1 - d)*(1 - d^2)*...*(1 - d^n) * (1 - u*d)*(1 - u*d^2)*...*(1 - u*d^n) ).

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

A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

Original entry on oeis.org

1, 1, 4, 5, 11, 22, 36, 69, 121, 221, 386, 686, 1210, 2122, 3734, 6517, 11408, 19903, 34714, 60485, 105312, 183272, 318758, 554262, 963361, 1674076, 2908426, 5052066, 8774386, 15237482, 26458718, 45939797, 79759442, 138468656, 240382216, 417289619, 724369536, 1257396992

Offset: 1

Paul D. Hanna, Jul 17 2013

nonn

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 11*x^5/5 + 22*x^6/6 +...
such L(x) = log(P(x)) - log(Q(x)) where
P(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...+ A224898(n)*x^n +...
Q(x) = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 - 2*x^15 - x^16 - 3*x^17 - 3*x^18 +...+ A039924(n)*x^n +...
log(P(x)) = -2*x^2/2 - 3*x^3/3 - 6*x^4/4 - 10*x^5/5 - 11*x^6/6 - 21*x^7/7 - 22*x^8/8 - 39*x^9/9 - 42*x^10/10 +...
log(Q(x)) = -x - 3*x^2/2 - 7*x^3/3 - 11*x^4/4 - 21*x^5/5 - 33*x^6/6 - 57*x^7/7 - 91*x^8/8 - 160*x^9/9 - 263*x^10/10 +...

Cf. A227543, A005169, A039924, A224898.

/* As the log of a continued fraction: */
{a(n)=local(A=x, CF=1+x); for(k=0, n, CF=1/(1-x^(n-k+1)*CF+x*O(x^n)); A=log(CF)); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=x, P=1+x, Q=1);
P=sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k));
Q=sum(m=0, sqrtint(n), (-1)^m*x^(m^2)/prod(k=1, m, 1-x^k));
A=log(P/(Q+x*O(x^n))); n*polcoeff(A, n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: log( 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(1-...)))))) ), the logarithm of a continued fraction.
L.g.f.: log( P(x) / Q(x) ) where
P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity.

A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24

Offset: 0

John Tyler Rascoe, Aug 26 2024

Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.

			Triangle begins:
    k=0  1  2   3   4   5   6   7   8   9  10  11  12  13  14
 n=0: 1;
 n=1: 1, 1, 1;
 n=2: 1, 2, 2,  2,  1,  1;
 n=3: 1, 3, 4,  5,  4,  4,  3,  2,  1,  1;
 n=4: 1, 4, 7, 10, 11, 11, 11,  9,  8,  6,  5,  3,  2,  1,  1;
 ...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).

John Tyler Rascoe, Python program.

Cf. A000245 (empirical row sums), A000217 (row lengths).
Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).
Cf. A000108, A152659, A173258, A227543, A268429.

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A366920 a(n) is the number times a Dyck path in an m X m box of any size has area n, counted to the lower right.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 5, 6, 7, 7, 5, 6, 8, 12, 15, 18, 16, 16, 15, 17, 24, 32, 40, 43, 45, 45, 42, 44, 53, 69, 87, 104, 115, 126, 125, 124, 124, 136, 160, 198, 240, 282, 321, 345, 360, 365, 367, 382, 417, 482, 574, 682, 791, 895, 976

Offset: 0

William J. Keith, Oct 28 2023

nonn

A Dyck path in an m X m grid is a set of up steps U and right steps R from the lower left corner to the upper right corner, staying weakly above the diagonal.
For this statistic, count the boxes below and to the right of the path.
The first time an area appears in two different squares is at size 15, which appears in the 4 X 4 box below UUURURRR and in the 5 X 5 box below URURURURUR.

			The 0 X 0 box yields the trivial (empty) path of area 0.
The 1 X 1 box yields one Dyck path of area 1 (UR).
The 2 X 2 box yields one Dyck path each of area 3 (URUR) and 4 (UURR).
The 3 X 3 box yields one Dyck path of area 6 (URURUR), two of area 7 (UURRUR and URUURR), and one each of area 8 (UURURR) and 9 (UUURRR).

Cf. A143951, A227543.

G.f.: 1 + q + q^3 + q^4 + q^6 + 2q^7 + ...

To construct the g.f., take A(x,q) as defined in A227543, and replace each instance of x^k with q^(k*(k+1)/2).

a(45)-a(65) from Alois P. Heinz, Oct 29 2023

cp's OEIS Frontend

A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

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A326453 Triangle read by rows: T(n,k) is the number of small Schröder paths of semilength k such that the area between the path and the x-axis is equal to n (n >= 0; 0 <= k <= n).

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A326454 Irregular triangle read by rows: T(n,k) is the number of small Schröder paths such that the area between the path and the x-axis is equal to n and contains k down-triangles.

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A274291 The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.

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A309086 Irregular triangle read by rows: T(n,k) is the number of small Schröder paths of semilength n such that the area between the path and the x-axis contains k down-triangles.

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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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A227620 Logarithmic derivative of A005169, the number of fountains of n coins.

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A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

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