A129182 -id:A129182 - cp's OEIS frontend

A143951 Number of Dyck paths such that the area between the x-axis and the path is n.

1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 71, 107, 161, 243, 367, 553, 834, 1258, 1898, 2863, 4318, 6514, 9827, 14824, 22361, 33732, 50886, 76762, 115796, 174680, 263509, 397508, 599647, 904579, 1364576, 2058489, 3105269, 4684359, 7066449, 10659877, 16080632, 24257950, 36593598, 55202165, 83273553, 125619799, 189499952

Offset: 0

Emeric Deutsch, Oct 09 2008

nonn

Column sums of A129182.

			a(5)=3 because we have UDUUDD, UUDDUD and UDUDUDUDUD, where U=(1,1) and D=(1,-1).
From _Peter Bala_, Dec 26 2012: (Start)
F(1/10) = sum {n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + ...)))))).
F(-1/10) = sum {n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(10 + 1/(100 + 1/(1000 + ...)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vincenzo Librandi)
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012

Cf. A129182, A291874 (convolution inverse).

Cf. A005169, A111317, A224704.

g:=1/(1-x/(1-x^3/(1-x^5/(1-x^7/(1-x^9/(1-x^11/(1-x^13/(1-x^15)))))))): gser:= series(g,x=0,45): seq(coeff(gser,x,n),n=0..44);
# second Maple program:
b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
      `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
    end:
a:= n-> add(b(2*n-4*t, 0, n), t=0..n/2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 24 2018

terms = 50; CoefficientList[1/(1+ContinuedFractionK[-x^(2i-1), 1, {i, 1, Sqrt[terms]//Ceiling}]) + O[x]^terms, x] (* Jean-François Alcover, Jul 11 2018 *)

N=66; q = 'q +O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / G(k+2) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013

G.f.: 1/(1 - x/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^9/(1 - ...
Derivation: the g.f. G(x,z) of Dyck paths, where x marks area and z marks semilength, satisfies G(x,z)=1+x*z*G(x,z)*G(x,x^2*z). Set z=1.
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^3-2 + 1/(1 + ...)))))).
For n >= 1, F(-1/n) has the simple continued fraction expansion
1/(1 + 1/(n + 1/(n^2 + 1/(n^3 + ...)))). Examples are given below. Cf. A005169 and A111317.
(End)
G.f.: A(x) = 1/(1 - x/(1-x + x/(1+x^2 + x^4/(1-x^3 - x^2/(1+x^4 - x^7/(1-x^5 + x^3/(1+x^6 + x^10/(1-x^7 - x^4/(1+x^8 - x^13/(1-x^9 + x^5/(1+x^10 + x^16/(1 + ...)))))))))))), a continued fraction. - Paul D. Hanna, Aug 08 2016
a(n) ~ c / r^n, where r = 0.66290148514884371255690407749133031115536799774051... and c = 0.337761150388539773466092171229604432776662930886727976914... . - Vaclav Kotesovec, Feb 17 2017, corrected Nov 04 2021
From Peter Bala, Jul 04 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ) and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2-n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ). Cf. A224704.
D(q) has its least  positive (and simple) real zero at x = 0.66290 14851 48843 71255 69040 ....
a(n) ~ c*d^n, where  d = 1/x = 1.5085197761707628638804960 ...  and c = - N(x)/(x*D'(x)) = 0.3377611503885397734660921 ... (the prime indicates differentiation w.r.t. q). (End)

b-file corrected and extended by Alois P. Heinz, Aug 24 2018

A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 1, 0, 6, 0, 6, 0, 1, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1, 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1, 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1, 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1, 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1

Offset: 0

Joerg Arndt, Mar 29 2014

Triangle A129182 transposed.
Column sums give the Catalan numbers (A000108).
Row sums give A143951.
Sums along falling diagonals give A005169.
T(4n,2n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			Triangle begins:
00:  1;
01:  0, 1;
02:  0, 0, 1;
03:  0, 0, 0, 1;
04:  0, 0, 1, 0, 1;
05:  0, 0, 0, 2, 0, 1;
06:  0, 0, 0, 0, 3, 0, 1;
07:  0, 0, 0, 1, 0, 4, 0, 1;
08:  0, 0, 0, 0, 3, 0, 5, 0, 1;
09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;
10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;
11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;
12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;
13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;
14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;
15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;
16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;
17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;
18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;
19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;
20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;
...
Column k=4 corresponds to the following 14 paths (dots denote zeros):
#:         path              area   steps (Dyck word)
01:  [ . 1 . 1 . 1 . 1 . ]     4     + - + - + - + -
02:  [ . 1 . 1 . 1 2 1 . ]     6     + - + - + + - -
03:  [ . 1 . 1 2 1 . 1 . ]     6     + - + + - - + -
04:  [ . 1 . 1 2 1 2 1 . ]     8     + - + + - + - -
05:  [ . 1 . 1 2 3 2 1 . ]    10     + - + + + - - -
06:  [ . 1 2 1 . 1 . 1 . ]     6     + + - - + - + -
07:  [ . 1 2 1 . 1 2 1 . ]     8     + + - - + + - -
08:  [ . 1 2 1 2 1 . 1 . ]     8     + + - + - - + -
09:  [ . 1 2 1 2 1 2 1 . ]    10     + + - + - + - -
10:  [ . 1 2 1 2 3 2 1 . ]    12     + + - + + - - -
11:  [ . 1 2 3 2 1 . 1 . ]    10     + + + - - - + -
12:  [ . 1 2 3 2 1 2 1 . ]    12     + + + - - + - -
13:  [ . 1 2 3 2 3 2 1 . ]    14     + + + - + - - -
14:  [ . 1 2 3 4 3 2 1 . ]    16     + + + + - - - -
There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is
[0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].
Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:
Semilength 4:
  [ . 1 . 1 2 1 2 1 . ]
  [ . 1 2 1 . 1 2 1 . ]
  [ . 1 2 1 2 1 . 1 . ]
Semilength 6:
  [ . 1 . 1 . 1 . 1 . 1 2 1 . ]
  [ . 1 . 1 . 1 . 1 2 1 . 1 . ]
  [ . 1 . 1 . 1 2 1 . 1 . 1 . ]
  [ . 1 . 1 2 1 . 1 . 1 . 1 . ]
  [ . 1 2 1 . 1 . 1 . 1 . 1 . ]
Semilength 8:
  [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))),  A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).
Cf. A047998, A138158, A227543.
Cf. A129181.

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),
       b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))
    end:
T:= (n, k)-> b(2*k, 0, n):
seq(seq(T(n, k), k=0..n), n=0..20);  # Alois P. Heinz, Mar 29 2014

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }
N=20; x='x+O('x^N);
F(x,y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
v= Vec( F(x,y) );
print_triangle(v)

G.f.: F(x,y) satisfies F(x,y) = 1 / (1 - x*y * F(x, x^2*y) ).

G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

A129176 Irregular triangle read by rows: T(n,k) is the number of Dyck words of length 2n having k inversions (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 3, 1, 1, 1, 2, 3, 5, 5, 7, 7, 6, 4, 1, 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 16, 17, 14, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 18, 22, 28, 32, 37, 40, 44, 43, 40, 35, 25, 15, 6, 1, 1, 1, 2, 3, 5, 7, 11, 15, 20, 26, 34, 42, 53, 63, 73, 85, 96, 106, 113, 118, 118, 115, 102, 86, 65, 41, 21, 7, 1

Offset: 0

Emeric Deutsch, Apr 11 2007

A Dyck word of length 2n is a word of n 0's and n 1's for which no initial segment contains more 1's than 0's.
Representing a Dyck word p of length 2n as a superdiagonal Dyck path p', the number of inversions of p is equal to the area between p' and the path that corresponds to the Dyck word 0^n 1^n.
Row n has 1+n(n-1)/2 terms. Row sums are the Catalan numbers (A000108). Alternating row sums for n>=1 are the Catalan numbers alternated with 0's (A097331). Sum(k*T(n,k),k>=0) = A029760(n-2).
This triangle is A129182 (area under Dyck paths), reflected and compressed (0's removed). Equivalently, A239927 rotated by Pi/2 clockwise and compressed.
This is also the number of Catalan paths of length n and area k. - N. J. A. Sloane, Nov 28 2011
From Alford Arnold, Jan 29 2008:
This triangle gives the partial sums of the following triangle A136624:
1
.1
....2...1
........2...3...3...1
............2...2...6...7...6...4...1
................2...2...4...8..12..15..17..14..10...5...1
etc.

			T(4,5) = 3 because we have 01010011, 01001101 and 00110101.
Triangle starts:
[0] 1;
[1] 1;
[2] 1, 1;
[3] 1, 1, 2, 1;
[4] 1, 1, 2, 3, 3, 3, 1;
[5] 1, 1, 2, 3, 5, 5, 7,  7,  6,  4,  1;
[6] 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 16, 17, 14, 10, 5, 1;
...

Alois P. Heinz, Rows n = 0..40, flattened
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, A, 40, 248-264, 1985.
M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.

Cf. A000108, A097331, A029760, A129182, A239927.
Cf. A136624, A136625.
Mirror image of A227543.

P[0]:=1: for n from 0 to 8 do
P[n+1]:=sort(expand(sum(t^((i+1)*(n-i))*P[i]*P[n-i],i=0..n))) od:
for n from 1 to 9 do seq(coeff(P[n],t,j),j=0..n*(n-1)/2) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, t)*z^t+b(x-1, y-1, t+1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 10 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, t]*z^t + b[x-1, y-1, t+1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

P(x, n) =
{
    if ( n<=1, return(1) );
    return( sum( i=0, n-1, P(x, i) * P(x, n-1 -i) * x^((i+1)*(n-1 -i)) ) );
}
for (n=0, 10, print( Vecrev( P(x,n) ) ) ); \\ Joerg Arndt, Jan 23 2024

\\ faster with memoization:
N=11;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
for (n=0, N, print( Vecrev( P(n) ) ) ); \\ Joerg Arndt, Jan 23 2024

from sage.combinat.q_analogues import qt_catalan_number
for n in (0..9): print(qt_catalan_number(n).substitute(q=1).coefficients())
# Peter Luschny, Mar 10 2020

The row generating polynomials P[n] = P[n](t) satisfy P[0] = 1 and

P[n+1] = Sum_{i=0..n} P[i] P[n-i] t^((i+1)*(n-i)).

A139262 Total number of two-element anti-chains over all ordered trees on n edges.

Original entry on oeis.org

0, 0, 1, 8, 47, 244, 1186, 5536, 25147, 112028, 491870, 2135440, 9188406, 39249768, 166656772, 704069248, 2961699667, 12412521388, 51854046982, 216013684528, 897632738722, 3721813363288, 15401045060572, 63616796642368, 262357557683422, 1080387930269464, 4443100381114156

Offset: 0

Lifoma Salaam, Apr 12 2008

nonn

From Miklos Bona, Mar 04 2009: (Start)
This is the same as the total number of inversions in all 132-avoiding permutations of length n by the well-known bijection between ordered trees on n edges and such permutations.
For example, there are five permutations of length three that avoid 132, namely, 123, 213, 231, 312, and 321. Their numbers of inversions are, respectively, 0,1,2,2, and 3, for a total of eight inversions.
(End)
Appears to be a shifted version of A029760. - R. J. Mathar, Mar 30 2014
a(n) is the number of total East steps below y = x-1 of all North-East paths from (0,0) to (n,n). Details can be found in Section 3.1 and Section 5 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

			a(3) = 8 because there are 5 ordered trees on 3 edges and two of the trees have 2 two-element anti-chain each, one of the trees has three two element anti-chains, one of the trees has one two element anti-chain and the last tree does not have any two-element anti-chains. Hence in ordered trees on 3 edges there are a total of (2)(2)+1(3)+1(1) = 8 two element anti-chains.

Robert Israel, Table of n, a(n) for n = 0..1650
Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), #P62, eq. (2).
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Lifoma Salaam, Combinatorial statistics on phylogenetic trees, Ph.D. Dissertation, Howard University, Washington D.C., 2008.
Sittipong Thamrongpairoj, Dowling Set Partitions, and Positional Marked Patterns, Ph. D. Dissertation, University of California-San Diego (2019).

Cf. A129182, A139263.

0, seq((n+1)*(2*n-1)!/(n!*(n-1)!) - 2^(2*n-1), n=1..30); # Robert Israel, Feb 02 2016

a[0] = 0; a[n_] := (n+1)(2n-1)!/(n! (n-1)!) - 2^(2n-1);
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 19 2018, from Maple *)

G.f.: (up to offset): A = x^2*(B^3)*(C^2), where B is the generating function for the central binomial coefficients and C is the generating function for the Catalan numbers. Thus A = x^2*(1/sqrt(1-4*x))^3*((1-sqrt(1-4*x))/2*x)^2.
2*a(n) = (n+1)*A000984(n) - 4^n. - J. M. Bergot, Feb 02 2013
Conjecture: n*(n-2)*a(n) +2*(-4*n^2+9*n-3)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 03 2013
The above conjecture follows easily from the formula by J. M. Bergot. - Robert Israel, Feb 02 2016
a(n) = Sum_{k=0..n^2} (n^2-k)/2 * A129182(n,k). - Alois P. Heinz, Mar 31 2018

Terms beyond a(9) added by Joerg Arndt, Dec 30 2012

A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 0, 7, 0, 5, 1, 2, 3, 6, 7, 8, 6, 6, 3, 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7, 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4, 2, 0, 11, 0, 29, 0, 63, 0, 115, 0, 176, 0, 238, 0, 255, 0, 230, 0, 169, 0, 92, 0, 41, 0, 9

Offset: 0

Alois P. Heinz, Mar 16 2018

			              .______.
              | /\/\ |  ,  rectangle area: 12, above path area: 5,
T(3,-1) = 1:  |/____\|  ,  below path area: 7, difference: (5-7) = 2 * (-1).
.
                 /\
                /  \
T(3,0) = 2:    /    \   /\/\/\  .
.
                /\         /\
T(3,1) = 2:    /  \/\   /\/  \  .
.
Triangle T(n,k) begins:
:                                   1                                    ;
:                                   1                                    ;
:                                   2                                    ;
:                               1,  2,  2                                ;
:                           2,  0,  7,  0,  5                            ;
:                   1,  2,  3,  6,  7,  8,  6,  6,  3                    ;
:           2,  0,  9,  0, 20,  0, 35,  0, 34,  0, 25,  0,  7            ;
:  1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4  ;

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Counting lattice paths

Row sums give A000108.
Column k=0 gives A300952.
Cf. A002620, A026741, A040001, A129182, A239927, A300322, A300996.

Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).
T(n,-floor((n-1)^2/4)) = A040001(n).
T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.
T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).

A240008 Number of Dyck paths of semilength 2n such that the area between the x-axis and the path is 4n.

Original entry on oeis.org

1, 1, 3, 14, 65, 301, 1419, 6786, 32749, 159108, 777224, 3813745, 18783934, 92811389, 459832745, 2283628771, 11364500644, 56659024320, 282939657220, 1414980598167, 7085590965083, 35523567248527, 178289298823240, 895697952270827, 4503912366189604

Offset: 0

Alois P. Heinz, Mar 30 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
      `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
    end:
a:= n-> b(4*n, 0, 4*n):
seq(a(n), n=0..30);

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0 || k>x^2/2-(y-x)^2/4, 0, If[x==0, 1, b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]];
a[n_] := b[4n, 0, 4n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)

a(n) = A129182(2n,4n) = A239927(4n,2n).

a(n) ~ c * d^n / sqrt(n), where d = 5.134082940807122222912767966569622... and c = 0.198313337349936555418443931967... - Vaclav Kotesovec, Apr 01 2014

A300322 Number T(n,k) of Dyck paths of semilength n such that 2k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 3, 6, 3, 1, 2, 5, 8, 12, 8, 5, 2, 1, 4, 9, 16, 22, 28, 22, 16, 9, 4, 1, 1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1, 2, 7, 15, 31, 53, 82, 114, 147, 171, 186, 171, 147, 114, 82, 53, 31, 15, 7, 2, 1, 5, 13, 30, 56, 95, 150, 216, 293, 371, 445, 495, 522, 495, 445, 371, 293, 216, 150, 95, 56, 30, 13, 5, 1

Offset: 0

Alois P. Heinz, Mar 02 2018

			               /\
T(3,-1) = 1:  /  \/\
.
                /\
               /  \     /\/\
T(3,0) = 3:   /    \   /    \   /\/\/\
.
                 /\
T(3,1) = 1:   /\/  \
.
Triangle T(n,k) begins:
:                             1                            ;
:                             1                            ;
:                             2                            ;
:                         1,  3,  1                        ;
:                     1,  3,  6,  3,  1                    ;
:                 2,  5,  8, 12,  8,  5,  2                ;
:         1,  4,  9, 16, 22, 28, 22, 16,  9,  4,  1        ;
:  1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1  ;

Alois P. Heinz, Rows n = 0..60, flattened
Wikipedia, Counting lattice paths

Row sums give A000108.
Column k=0 gives A300323.
Cf. A130518, A129182, A177702, A239927, A298645, A300953.

b:= proc(x, y, v) option remember; expand(
     `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
      b(x-1, y+i, v+j)*z^((y-v)/2+(i-j)/4), i=l), j=l))([-1, 1]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=ldegree(p)..degree(p)))(
             add(b(n, (n-2*j)$2), j=0..n/2)):
seq(T(n), n=0..12);

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[b[x - 1, y + i, v + j] z^((y - v)/2 + (i - j)/4), {i, l}], {j, l}]][{-1, 1}]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, Range[Exponent[p, z, Reverse @@ # &], Exponent[p, z]]}]][Sum[b[n, n-2j, n-2j], {j, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 31 2018, from Maple *)

T(n,k) = T(n,-k).

T(n,A130518(n)) = A177702(n).

A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 28, 69, 186, 522, 1536, 4638, 14408, 45568, 146884, 479871, 1589516, 5320854, 18000198, 61412376, 211282386, 731973720, 2553168136, 8957554412, 31604599044, 112060048354, 399227283950, 1428315878002, 5130964125124, 18499652813682

Offset: 0

Alois P. Heinz, Mar 02 2018

nonn

			              /\
             /  \      /\/\
a(3) = 3:   /    \    /    \    /\/\/\ .
.
a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths:
             /\  /\
          /\/  \/  \  and its reversal.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Counting lattice paths

Column k=0 of A300322.

Cf. A000108 (all Dyck paths), A000225, A001405 (symmetric Dyck paths), A129182, A239927, A298645.

b:= proc(x, y) option remember; expand(`if`(x=0, 1,
      `if`(y<1,   0, b(x-1, y-1)*z^(2*y-1))+
      `if`(x add(coeff(p, z, i)^2
      , i=0..degree(p)))(b(n, n-2*j)), j=0..n/2)
    end:
seq(a(n), n=0..32);

b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]];
a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) >= A001405(n) with equality only for n <= 4.

a(n) is odd <=> n in { A000225 }.

cp's OEIS Frontend

A143951 Number of Dyck paths such that the area between the x-axis and the path is n.

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A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

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A129176 Irregular triangle read by rows: T(n,k) is the number of Dyck words of length 2n having k inversions (n >= 0, k >= 0).

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A139262 Total number of two-element anti-chains over all ordered trees on n edges.

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A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

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A240008 Number of Dyck paths of semilength 2n such that the area between the x-axis and the path is 4n.

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A300322 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n*(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.

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A300322 Number T(n,k) of Dyck paths of semilength n such that 2k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.