A052709 -id:A052709 - cp's OEIS frontend

A350252 Number of non-alternating patterns of length n.

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A063020 Reversion of y - y^2 - y^3 + y^4.

Original entry on oeis.org

0, 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 611871, 2685732, 11896906, 53115412, 238767737, 1079780412, 4909067468, 22424085244, 102865595140, 473678981820, 2188774576575, 10145798119530, 47165267330415, 219839845852692, 1027183096151244, 4810235214490986

Offset: 0

Olivier Gérard, Jul 05 2001

Seems to be the inverse of A007858. Can someone prove this?

a(n+1) counts paths from (0,0) to (n,n) which do not go above the line y=x, using steps (1,0) and (2k,1), where k ranges over the nonnegative integers. For example, the 9 paths from (0,0) to (3,3) are the 5 Catalan paths, as well as DNEN, DENN, EDNN and ENDN. Here E=(1,0), N=(0,1), D=(2,1). - Brian Drake, Sep 20 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
A. Mironov and A. Morozov, Algebra of quantum C-polynomials, arXiv:2009.11641 [hep-th], 2020.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
Index entries for reversions of series

Cf. A007848, A052709, A064641, A348410.

A:= series(RootOf(Z-_Z^2-_Z^3+_Z^4-x), x, 21): seq(coeff(A,x,i), i=0..20); # _Brian Drake, Sep 20 2007

CoefficientList[InverseSeries[Series[y - y^2 - y^3 + y^4, {y, 0, 30}], x], x]

a(n):=sum((sum(binomial(j,n-3*k+2*j-1)*(-1)^(j-k)*binomial(k,j),j,0,k))*binomial(n+k-1,n-1),k,0,n-1)/n; /* Vladimir Kruchinin, Oct 11 2011 */

a(n):=sum((-1)^(i)*binomial(n+i-1,i)*binomial(3*n-i-2,n-i-1),i,0,n-1)/n; /* Vladimir Kruchinin, Feb 13 2014 */

x='x+O('x^66); concat([0],Vec(serreverse(x-x^2-x^3+x^4))) \\ Joerg Arndt, May 28 2013

def b(n):
    h = binomial(3*n + 1, n) * hypergeometric([-n, n + 1], [-3*n - 1], -1) / (n + 1)
    return simplify(h)
print([0] + [b(n) for n in range(27)])  # Peter Luschny, Sep 21 2023

a(n) = (1/n)*Sum_{k=0..n-1} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-3*k+2*j-1)*(-1)^(j-k)*binomial(k,j). - Vladimir Kruchinin, Oct 11 2011
a(n) = (1/n)*Sum_{i=0..n-1} (-1)^(i)*binomial(n+i-1,i)*binomial(3*n-i-2,n-i-1), n > 0. - Vladimir Kruchinin, Feb 13 2014
Recurrence: 16*(n-1)*n*(2*n-1)*(17*n-27)*a(n) = (n-1)*(1819*n^3 - 6527*n^2 + 7350*n - 2520)*a(n-1) + 8*(2*n-3)*(4*n-9)*(4*n-7)*(17*n-10)*a(n-2). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ sqrt(11-3/sqrt(17))/16 * (107+51*sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(6*n)). - Vaclav Kotesovec, Feb 13 2014
The g.f. A(x) satisfies x*A'(x)/A(x) = 1 + x + 5*x^2 + 19*x^3 + 85*x^4 + ..., the g.f. of A348410. - Peter Bala, Feb 22 2022

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/( ... )))))).

A350354 Number of up/down (or down/up) patterns of length n.

Original entry on oeis.org

1, 1, 1, 3, 11, 51, 281, 1809, 13293, 109899, 1009343, 10196895, 112375149, 1341625041, 17249416717, 237618939975, 3491542594727, 54510993341523, 901106621474801, 15723571927404189, 288804851413993941, 5569918636750820751, 112537773142244706427

Offset: 0

Gus Wiseman, Jan 16 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A patten is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase.
A pattern is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).
Conjecture: Also the half the number of weakly up/down patterns of length n.
These are the values of the Euler zig-zag polynomials A205497 evaluated at x = 1/2 and normalized by 2^n. - Peter Luschny, Jun 03 2024

			The a(0) = 1 through a(4) = 11 patterns:
  ()  (1)  (1,2)  (1,2,1)  (1,2,1,2)
                  (1,3,2)  (1,2,1,3)
                  (2,3,1)  (1,3,1,2)
                           (1,3,2,3)
                           (1,3,2,4)
                           (1,4,2,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,4,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for permutations is A000111, undirected A001250.
For compositions we have A025048, down/up A025049, undirected A025047.
This is the up/down (or down/up) case of A345194.
A205497 are the Euler zig-zag polynomials.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A226316 counts patterns avoiding (1,2,3), weakly A052709.
A335515 counts patterns matching (1,2,3).
A349058 counts weakly alternating patterns.
A350252 counts non-alternating patterns.
Row sums of A079502.
Cf. A000041, A003242, A049774, A124754, A128761, A335457, A344604, A344605, A344615, A348610.

# Using the recurrence by Kyle Petersen from A205497.
G := proc(n) option remember; local F;
if n = 0 then 1/(1 - q*x) else F := G(n - 1);
simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
A350354 := n -> 2^n*subs({p = 1, q = 1, x = 1/2}, G(n)*(1 - x)^(n + 1)):
seq(A350354(n), n = 0..22);  # Peter Luschny, Jun 03 2024

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, 0, F(k-2,-x)/F(k-1,x)-1) + x + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n > 2) = A344605(n)/2.

a(n > 1) = A345194(n)/2.

Terms a(10) and beyond from Andrew Howroyd, Feb 04 2022

A366273 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^5.

Original entry on oeis.org

1, 1, 9, 81, 849, 9681, 116601, 1459809, 18809121, 247782369, 3322209001, 45187029809, 621970864241, 8647376531249, 121261376439641, 1713085987837889, 24358211622230081, 348325689458584769, 5006342381846708681, 72279683684984063249, 1047789195353379807121

Offset: 0

Seiichi Manyama, Oct 06 2023

nonn

Cf. A052709, A366221, A366272.

Cf. A366268.

a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k,n-k) * binomial(5*k,k)/(4*k+1).

A071943 Triangle T(n,k) (n>=0, 0 <= k <= n) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R=(1,0), V=(0,1) and D=(1,2).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 1, 4, 12, 24, 31, 1, 5, 18, 46, 89, 113, 1, 6, 25, 76, 183, 342, 431, 1, 7, 33, 115, 323, 741, 1355, 1697, 1, 8, 42, 164, 520, 1376, 3054, 5492, 6847, 1, 9, 52, 224, 786, 2326, 5900, 12768, 22669, 28161, 1, 10, 63, 296, 1134, 3684, 10370

Offset: 0

N. J. A. Sloane, Jun 15 2002

For another interpretation of this array see the Example section.

			T(3,2)=7 because we have RRRVV, RRVRV, RRVVR, RVRRV, RVRVR, RRD and RDR.
Array begins:
1,
1, 1,
1, 2, 3,
1, 3, 7, 9,
1, 4, 12, 24, 31,
1, 5, 18, 46, 89, 113,
1, 6, 25, 76, 183, 342, 431,
1, 7, 33, 115, 323, 741, 1355, 1697,
...
Equivalently, let U(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,-1) and (0,-1). Then U(n,n-k) = T(n,k). The U(n,k) array begins:
4:  0  0  0  0  1  5 ...
3:  0  0  0  1  4 18 ...
2:  0  0  1  3 12 46 ...
1:  0  1  2  7 24 89 ...
0:  1  1  3  9 31 113 ...
-------------------------
k/n:0  1  2  3  4  5 ...
The recurrence for this version is: U(0,0)=1, U(n,k)=0 for k>n or k<0; U(n,k) = U(n,k+1) + U(n-1,k+1) + U(n,k-1). E.g. 46 = 18 + 4 + 24. Also U(n,0) = A052709(n-1).

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
N. J. A. Sloane, Rows 0 through 100
N. J. A. Sloane, Illustration of the initial terms of the U(n,k) array

Diagonal entries yield A052709. Row sums are A071356.

Related arrays: A071944, A071945, A071946.

U:=proc(n,k) option remember;
if (n < 0) then RETURN(0);
elif (n=0) then
   if (k=0) then RETURN(1); else RETURN(0); fi;
elif (k>n or k<0) then RETURN(0);
else RETURN(U(n,k+1)+U(n-1,k+1)+U(n-1,k-1));
fi;
end;
for n from 0 to 20 do
lprint( [seq(U(n,n-i),i=0..n)] );
od:

t[0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, k] + t[n-1, k-2]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after N. J. A. Sloane *)

G.f.=(1-q)/[z(2t+2t^2z-1+q)], where q=sqrt(1-4tz-4t^2z^2).

Define T(0,0)=1 and T(n,k)=0 for k<0 and k >n. Then the array is generated by the recurrence T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-2). For example, T(5,3) = 46 = T(5,2) + T(4,3) + T(4,1) = 18 + 24 + 4. - N. J. A. Sloane, Mar 28 2013

Edited by Emeric Deutsch, Dec 21 2003

Edited by N. J. A. Sloane, Mar 28 2013

A071945 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using steps R=(1,0), V=(0,1) and D=(2,1).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 31, 31, 1, 9, 33, 73, 113, 113, 1, 11, 51, 143, 287, 431, 431, 1, 13, 73, 249, 609, 1153, 1697, 1697, 1, 15, 99, 399, 1151, 2591, 4719, 6847, 6847, 1, 17, 129, 601, 2001, 5201, 11073, 19617, 28161, 28161, 1, 19, 163, 863, 3263

Offset: 0

N. J. A. Sloane, Jun 15 2002

Also could be titled: "Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess king from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves." - Peter Kagey, Apr 20 2020

			a(3,1)=5 because we have RRRV, RRVR, RVRR, RD and DR.
Triangle begins:
1
1 1
1 3    3
1 5    9   9
1 7   19  31   31
1 9   33  73  113  113
1 11  51 143  287  431   431
1 13  73 249  609 1153  1697  1697
1 15  99 399 1151 2591  4719  6847  6847
1 17 129 601 2001 5201 11073 19617 28161 28161

Peter Kagey, Table of n, a(n) for n = 0..8255 (first 128 rows, flattened)
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Diagonal entries give A052709.

G.f.: (1-q)/[z(1+tz)(2t-1+q)], where q=sqrt(1-4tz-4t^2z^2).

Edited by Emeric Deutsch, Dec 21 2003

A360076 a(n) = Sum_{k=0..n} binomial(3k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 5, 20, 90, 430, 2136, 10937, 57307, 305822, 1656482, 9083432, 50328114, 281324294, 1584578746, 8984740485, 51242962251, 293772468164, 1691974930584, 9785378133297, 56805049768157, 330880419984832, 1933299689139364, 11328101469158554

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Cf. A052709, A073155, A360082, A360083.

Cf. A000108, A099234.

a(n) = sum(k=0, n, binomial(3*k, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^3)))

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^3 * A(x)).

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^3 )).

A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 4, 5, 0, 0, 2, 15, 14, 0, 0, 0, 15, 56, 42, 0, 0, 0, 5, 84, 210, 132, 0, 0, 0, 0, 56, 420, 792, 429, 0, 0, 0, 0, 14, 420, 1980, 3003, 1430, 0, 0, 0, 0, 0, 210, 2640, 9009, 11440, 4862, 0, 0, 0, 0, 0, 42, 1980, 15015, 40040, 43758, 16796, 0, 0, 0, 0, 0, 0, 792, 15015, 80080, 175032, 167960, 58786

Offset: 0

Paul Barry, Mar 14 2006

			Triangle begins as:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 4,  5;
  0, 0, 2, 15, 14;
  0, 0, 0, 15, 56,  42;
  0, 0, 0,  5, 84, 210, 132;
  0, 0, 0,  0, 56, 420, 792, 429;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A052709, A115178, A115178.

[Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021

Table[CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 31 2021 *)

flatten([[binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021

T(n, k) = binomial(k, n-k)*Catalan(k).
Sum_{k=0..n} T(n, k) = A052709(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115178(n) (upward diagonal sums).
T(n, k) = (-1)^(n+k)*A115179(n, k).

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 6, 27, 134, 709, 3892, 22004, 127250, 749230, 4476386, 27071344, 165398868, 1019405720, 6330482488, 39571612357, 248796862550, 1572300095758, 9981970108384, 63633339713190, 407162295120570, 2614059813642256, 16834457481559076

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Cf. A052709, A073155, A360076, A360083.

Cf. A000108, A099235.

a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^4)))

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^4 * A(x)).

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^4 )).

cp's OEIS Frontend

A350252 Number of non-alternating patterns of length n.

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A063020 Reversion of y - y^2 - y^3 + y^4.

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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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A350354 Number of up/down (or down/up) patterns of length n.

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A366273 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^5.

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A071943 Triangle T(n,k) (n>=0, 0 <= k <= n) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R=(1,0), V=(0,1) and D=(1,2).

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A071945 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using steps R=(1,0), V=(0,1) and D=(2,1).

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A366273 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^5.

A360076 a(n) = Sum_{k=0..n} binomial(3k,n-k) Catalan(k).

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).