A002694 -id:A002694 - cp's OEIS frontend

A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

1, 2, 1, 5, 6, 2, 14, 28, 20, 5, 42, 120, 135, 70, 14, 132, 495, 770, 616, 252, 42, 429, 2002, 4004, 4368, 2730, 924, 132, 1430, 8008, 19656, 27300, 23100, 11880, 3432, 429, 4862, 31824, 92820, 157080, 168300, 116688, 51051, 12870, 1430

Offset: 0

N. J. A. Sloane, Jan 11 2014

			Triangle begins:
     1,
     2,    1,
     5,    6,     2,
    14,   28,    20,     5,
    42,  120,   135,    70,    14,
   132,  495,   770,   616,   252,    42,
   429, 2002,  4004,  4368,  2730,   924,  132,
  1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
  ...

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened
Antoine Abram, Florent Hivert, James D. Mitchell, Jean-Christophe Novelli, and Maria Tsalakou, Power Quotients of Plactic-like Monoids, arXiv:2406.16387 [math.CO], 2024. See p. 5.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.
Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2. See pp. 21-22. - _N. J. A. Sloane_, Jul 12 2014

A062991 is a signed version. See also A094385 for another version.
Cf. A009766.
The two borders give the Catalan numbers A000108.
Cf. A062992 (row sums).
The second and third columns give A002694 and A244887.

a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
a234950_row n = map (a234950 n) [0..n]
a234950_tabl = map a234950_row [0..]
-- Reinhard Zumkeller, Jan 12 2014

T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Sep 04 2018

T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Sep 06 2015

G.f.: 1/x*(1-sqrt(1-4*x-4*x*y))/(1+2*y+sqrt(1-4*x-4*x*y)). - Vladimir Kruchinin, Sep 04 2018

T(n,k) = 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)). - Peter Luschny, Sep 04 2018

A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2x + 1)^n + (x^2 - 1)(x + 1)^n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 2, 6, 6, 2, 0, 3, 13, 22, 18, 7, 1, 0, 4, 23, 56, 75, 60, 29, 8, 1, 0, 5, 36, 115, 215, 261, 215, 121, 45, 10, 1, 0, 6, 52, 206, 495, 806, 938, 798, 496, 220, 66, 12, 1, 0, 7, 71, 336, 987, 2016, 3031, 3452, 3010, 2003, 1001, 364, 91

Offset: 0

Franck Maminirina Ramaharo, Feb 28 2018

			The triangle T(n, k) begins:
n\k  0  1   2    3    4     5     6     7     8     9    10   11  12  13 14
0:   0  0   1
1:   0  1   2    1
2:   0  2   6    6    2
3:   0  3  13   22   18     7     1
4:   0  4  23   56   75    60    29     8     1
5:   0  5  36  115  215   261   215   121    45    10     1
6:   0  6  52  206  495   806   938   798   496   220    66   12   1
7:   0  7  71  336  987  2016  3031  3452  3010  2003  1001  364  91  14  1

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996.

Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
A. M. Mathai and P. N. Rathie, Enumeration of almost cubic maps, Journal of Combinatorial Theory, Series B, Vol 13 (1972), 83-90.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row sums: A000302 (powers of 4).

Cf. A034870, A034871, A032443.

T := (n, k) -> binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k);
for n from 0 to 10 do seq(T(n, k), k = 0 .. max(2*n, n + 2)) od;

T(n, k) := binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k)$
a : []$
for n:0 thru 10 do
  a : append(a, makelist(T(n, k), k, 0, max(2*n, n + 2)))$
a;

row(n) = Vecrev((x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n); \\ Michel Marcus, Nov 12 2022

T(n,k) = binomial(2*n,k) + binomial(n,k-2) - binomial(n,k).
T(n,k) = T(n-1,k-1)+ T(n-1,k) + A034871(n-1,k-1), with T(n,0) = T(0,1) = 0 and T(0,2) = 1
T(n,1) = A001477(n).
T(n,2) = A143689(n).
T(n,3) = n + A002492(n-1) - A000292(n-2).
T(n,n) = A247493(n+1,n).
T(n,n+1) = n + A001791(n).
T(n,n+2) = 1 + A002694(n), n >= 2.
T(n,n+k) = binomial(2*n, n-k) = A094527(n,k), for k >= 3 and n>=k.
G.f.: 1/(1 - y*(x^2 + 2*x + 1)) + (x^2 - 1)/(1 - y*(x + 1)).

A004334 Binomial coefficient C(4n,n-4).

Original entry on oeis.org

1, 20, 276, 3276, 35960, 376992, 3838380, 38320568, 377348994, 3679075400, 35607051480, 342700125300, 3284214703056, 31368725759168, 298824321028320, 2840671544105280, 26958221130508525, 255485622301674660

Offset: 4

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Cf. binomial(k*n, n-k): A000027 (k=1), A002694 (k=2), A004321 (k=3), this sequence (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).

List([4..30], n-> Binomial(4*n,n-4)); # G. C. Greubel, Mar 21 2019

[Binomial(4*n, n-4): n in [4..30]]; // Vincenzo Librandi, Feb 01 2017

Table[Binomial[4n, n-4], {n,4,30}] (* Vincenzo Librandi, Feb 01 2017 *)

a(n)=binomial(4*n,n-4) \\ Charles R Greathouse IV, Feb 01 2017

[binomial(4*n,n-4) for n in (4..30)] # G. C. Greubel, Mar 21 2019

From Ilya Gutkovskiy, Jan 31 2017: (Start)
E.g.f.: (1/24)*x^4*3F3(17/4,9/2,19/4; 17/3,6,19/3; 256*x/27).
a(n) ~ 2^(8*n+1/2)/(sqrt(Pi*n)*3^(3*n+9/2)). (End)
D-finite with recurrence -3*(3*n+2)*(n-4)*(3*n+4)*(n+1)*a(n) +8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

A006659 Number of closed meander systems of order n+1 with n components.

Original entry on oeis.org

2, 12, 56, 240, 990, 4004, 16016, 63648, 251940, 994840, 3922512, 15452320, 60843510, 239519700, 942871200, 3711935040, 14615744220, 57562286760, 226760523600, 893550621600, 3522078700140, 13887053160552

Offset: 1

D. Ivanov, S. K. Lando and A. K. Zvonkin (zvonkin(AT)labri.u-bordeaux.fr)

a(n) is the total number of long interior inclines in all Dyck (n+2)-paths. An incline is a maximal subpath of like steps (all Us or all Ds); interior means it does not start or end the path; long means of length >= 2. Example: for n=1, the 5 Dyck 3-paths are shown with long interior inclines in uppercase: uuuddd, uududd, udUUdd, ududud, uuDDud and so a(1)=2. - David Callan, Jul 03 2006

a(n) is the number of corners in all parallelogram polyominoes of semiperimeter n+3. - Emeric Deutsch, Oct 09 2008

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Delest, J. P. Dubernard and I. Dutour, Parallelogram polyominoes and corners, J. Symbolic Computation, 20(1995),503-515.
M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3 (1987), 325-339.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis UQAM 1992, arXiv:0911.4975 [math.NT], 2009.

Cf. A002057, A001622.
Equals 2*A002694(n+1).
Cf. A005315, A005316.
A diagonal of triangle A008828.

a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
-- Reinhard Zumkeller, Jun 18 2012

seq(2*binomial(2*n+2,n-1),n=1..22); # Emeric Deutsch, Oct 09 2008

f[x_] := 32/((1 + Sqrt[1 - 4x])^4*Sqrt[1 - 4x]); CoefficientList[ Series[ f[x], {x, 0, 21}], x] (* Jean-François Alcover, Dec 07 2011 *)
CoefficientList[Series[4*Exp[2x](BesselI[1,2*x]+ x(x-1)(BesselI[0,2x]+BesselI[1,2x]))/x^2,{x,0,22}],x]Table[n!,{n,0,22}] (* Stefano Spezia, May 10 2022 *)

a(n)=2*binomial(2*n+2,n-1) \\ Charles R Greathouse IV, Dec 07 2011

x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ Altug Alkan, Oct 14 2015

G.f.: 32/(sqrt(1-4x)*(1+sqrt(1-4x))^4).
a(n) = (n+1) * A002057(n). - Ralf Stephan, Aug 31 2003
a(n) = 2*binomial(2n+2, n-1). - Emeric Deutsch, Oct 09 2008
a(n) = {(-56 - 30*n - 4*n^2)*a(n+1) + (8*n+12+n^2)*a(n+2), a(0)=2, a(1)=12}. - Simon Plouffe (master's thesis, 1992)
a(n) ~ 2^(2*n+3)/sqrt(n*Pi). - Charles R Greathouse IV, Dec 07 2011
E.g.f.: 4*exp(2*x)*(I_1(2*x) + x*(x - 1)*(I_0(2*x) + I_1(2*x)))/x^2, where I_n(x) is the modified Bessel function of the first kind. - Stefano Spezia, May 09 2022
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 23/12 - 13*Pi/(18*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 53*log(phi)/(5*sqrt(5)) - 37/20, where phi is the golden ratio (A001622). (End)

A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 36, 6, 105, 26, 1, 317, 104, 8, 982, 402, 45, 1, 3105, 1522, 225, 10, 9981, 5693, 1052, 69, 1, 32520, 21144, 4698, 412, 12, 107157, 78188, 20319, 2249, 98, 1, 356481, 288340, 85864, 11522, 679, 14, 1195662, 1061520, 356535, 56360, 4230

Offset: 0

Emeric Deutsch, Nov 29 2005

Row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k),k=0..floor(n/2)-1)=binomial(2n-4,n) (A002694). Column 0 yields A114465.

			T(5,1) = 6 because we have UUD(UU)DUDDD, UUD(UU)DDUDD, UUD(UU)DDDUD,
UDUUD(UU)DDD, UUDUD(UU)DDD and UUUDD(UU)DDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an odd level are shown between parentheses; note that the fourth path has an ascent of length 2 that starts at an even level).
Triangle starts:
:  0 :    1;
:  1 :    1;
:  2 :    2;
:  3 :    5;
:  4 :   13,    1;
:  5 :   36,    6;
:  6 :  105,   26,    1;
:  7 :  317,  104,    8;
:  8 :  982,  402,   45,  1;
:  9 : 3105, 1522,  225, 10;
: 10 : 9981, 5693, 1052, 69, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A114462, A114464, A114465, A000108, A002694, A243752.

G:=-1/2*(1-z^2+z^2*t-sqrt((z^2*t-z^2+4*z-1)*(z^2*t-z^2-1)))/z/(-z^2+z^2*t+z-z*t-1): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 2 to 15 do seq(coeff(t*P[n],t^j),j=1..floor(n/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, [2, 2, 2, 5, 2][t])
      *`if`(t=5, z, 1) +b(x-1, y-1, [1, 3, 4, 1, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # 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, {2, 2, 2, 5, 2}[[t]]]*If[t==5, z, 1] + b[x-1, y-1, {1, 3, 4, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

G.f.: G=G(t, z) satisfies z[(1-t)z^2-(1-t)z+1]G^2-[1-(1-t)z^2]G+1=0.

A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 6, 42, 28, 3, 132, 120, 28, 1, 429, 495, 180, 20, 1430, 2002, 990, 195, 10, 4862, 8008, 5005, 1430, 165, 4, 16796, 31824, 24024, 9009, 1650, 117, 1, 58786, 125970, 111384, 51688, 13013, 1617, 70, 208012, 497420, 503880, 278460, 89180, 16016, 1386, 35

Offset: 0

Alois P. Heinz, Jun 20 2016

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

			              /\
              \ \
T(3,1) = 1:   /  \
.
Triangle T(n,k) begins:
:     1;
:     1;
:     2;
:     5,     1;
:    14,     6;
:    42,    28,     3;
:   132,   120,    28,    1;
:   429,   495,   180,   20;
:  1430,  2002,   990,  195,   10;
:  4862,  8008,  5005, 1430,  165,   4;
: 16796, 31824, 24024, 9009, 1650, 117, 1;

Alois P. Heinz, Rows n = 0..160, flattened

Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).
Row sums give A230823.
Last elements of rows give A092392(n-1) for n>0.
Cf. A083920, A274405.

b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
      `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
      `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(0$3, 2*n)):
seq(T(n), n=0..14);

b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y > n, 0,
     If[n == y, If[t == 2, 0, 1], b[x + 1, y + 1, 0, n - 1] +
     If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
     If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A274405(n).

A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 4, 6, 5, 0, 27, 36, 28, 14, 0, 248, 310, 225, 120, 42, 0, 2830, 3396, 2332, 1210, 495, 132, 0, 38232, 44604, 29302, 14560, 6006, 2002, 429, 0, 593859, 678696, 430200, 204540, 81900, 28392, 8008, 1430, 0, 10401712, 11701926, 7204821, 3289296, 1263780, 431256, 129948, 31824, 4862

Offset: 0

R. J. Mathar, Dec 06 2018

If all subsets are allowed instead of just pairs (chords), we get A324173. The rightmost column is A000108 (see Riordan). - Gus Wiseman, Feb 27 2019

			From _Gus Wiseman_, Feb 27 2019: (Start)
Triangle begins:
  1
  0      1
  0      1      2
  0      4      6      5
  0     27     36     28     14
  0    248    310    225    120     42
  0   2830   3396   2332   1210    495    132
  0  38232  44604  29302  14560   6006   2002    429
  0 593859 678696 430200 204540  81900  28392   8008   1430
Row n = 3 counts the following chord diagrams (see link for pictures):
  {{1,3},{2,5},{4,6}}  {{1,2},{3,5},{4,6}}  {{1,2},{3,4},{5,6}}
  {{1,4},{2,5},{3,6}}  {{1,3},{2,4},{5,6}}  {{1,2},{3,6},{4,5}}
  {{1,4},{2,6},{3,5}}  {{1,3},{2,6},{4,5}}  {{1,4},{2,3},{5,6}}
  {{1,5},{2,4},{3,6}}  {{1,5},{2,3},{4,6}}  {{1,6},{2,3},{4,5}}
                       {{1,5},{2,6},{3,4}}  {{1,6},{2,5},{3,4}}
                       {{1,6},{2,4},{3,5}}
(End)

P. Flajolet and M. Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, p 191-201, eq (2)
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Gus Wiseman, Chords diagrams with 3 chords, organized by number of components.

Cf. A000699 (k = 1 column), A001147 (row sums), A000108 (diagonal), A002694 (subdiagonal k = n - 1).

Cf. A000096, A003436, A016098, A099947, A136653, A278990, A293157, A324173, A324323, A324327, A324328.

The g.f. satisfies g(z,w) = 1+w*A000699(w*g^2), where A000699(z) is the g.f. of A000699.

Offset changed to 0 by Gus Wiseman, Feb 27 2019

A371964 a(n) is the sum of all symmetric valleys in the set of Catalan words of length n.

Original entry on oeis.org

0, 0, 0, 0, 1, 7, 35, 155, 650, 2652, 10660, 42484, 168454, 665874, 2627130, 10353290, 40775045, 160534895, 631970495, 2487938015, 9795810125, 38576953505, 151957215305, 598732526105, 2359771876175, 9303298456451, 36688955738099, 144732209103699, 571117191135799

Offset: 0

Stefano Spezia, Apr 14 2024

nonn

			a(4) = 1 because there is 1 Catalan word of length 4 with one symmetric valley: 0101.
a(5) = 7 because there are 7 Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212 (see example at p. 16 in Baril et al.).

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 Corollary 4.7, pp. 16-17.

Cf. A371963, A371965.

Cf. A000108, A002694, A079309.

a:= proc(n) option remember; `if`(n<4, 0,
      a(n-1)+binomial(2*n-4, n-4))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 15 2024

CoefficientList[Series[(1-4x+2x^2-(1-2x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,29}],x]

from math import comb
def A371964(n): return sum(comb((n-i<<1)-4,n-i-4) for i in range(n-3)) # Chai Wah Wu, Apr 15 2024

G.f.: (1 - 4*x + 2*x^2 - (1 - 2*x)*sqrt(1 - 4*x))/(2*(1 - x)*sqrt(1 - 4*x)).
a(n) = (3*n - 2)*A000108(n-1) - A079309(n) for n > 0.
a(n) ~ 2^(2*n)/(12*sqrt(Pi*n)).
a(n)/A371963(n) ~ 1/2.
a(n) - a(n-1) = A002694(n-2).

A074922 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 2 simple intersections.

Original entry on oeis.org

0, 0, 0, 3, 28, 180, 990, 5005, 24024, 111384, 503880, 2238390, 9806280, 42493880, 182530530, 778439025, 3300049200, 13919756400, 58462976880, 244639718730, 1020422356200, 4244365452600, 17610393500700, 72907029092898

Offset: 0

Henry Bottomley, Oct 06 2002

			a(3)=3 since the only possibility is to have one of the three chords intersected by the other two.

Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Vincent Pilaud and Juanjo Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440, 2013
Henry Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A067310, A002694, A232224, A274404.

Table[Binomial[2n,n-2] (n-2)/2,{n,0,30}] (* Harvey P. Dale, Nov 04 2011 *)

a(n) = C(2n, n-2)*(n-2)/2 = A002694(n)*(n-2)/2 = A067310(n, 2) = Sum_{0<=j

A213344 2-quantum transitions in systems of N>=2 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 6, 24, 4, 80, 40, 240, 240, 15, 672, 1120, 210, 1792, 4480, 1680, 56, 4608, 16128, 10080, 1008, 11520, 53760, 50400, 10080, 210, 28160, 168960, 221760, 73920, 4620, 67584, 506880, 887040, 443520, 55440, 792

Offset: 2

Stanislav Sykora, Jun 09 2012

For a general discussion, please see A213343.
This a(n) is for double-quantum transitions (q = 2).
It lists the flattened triangle T(2;N,k) with rows N = 2,3,... and columns N, k = 0..floor((N-2)/2).

			For N=4, there are 4 second-quantum transitions with combination index 1: (0001,1110),(0010,1101),(0100,1011),(1000,0111).
Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-2)/2)
  2 |   1
  3 |   6
  4 |  24   4
  5 |  80  40
  6 | 240 240 15

See A213343.

Stanislav Sykora, Table of n, a(n) for n = 2..2501
Stanislav Sykora, T(2;N,k) with rows N=2,..,100 and columns k=0,..,floor((N-2)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213345 to A213352 (q=3..10).

Cf. A001788 (first column), A002694 (row sums).

With[{q = 2}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 12}, {k, 0, Floor[(n - 2)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 2.
```

Set q = 2 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

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A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

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A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n.

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A004334 Binomial coefficient C(4n,n-4).

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A006659 Number of closed meander systems of order n+1 with n components.

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A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

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A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

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A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2x + 1)^n + (x^2 - 1)(x + 1)^n.