A106228 -id:A106228 - cp's OEIS frontend

A371724 G.f. satisfies A(x) = ( 1 + x * A(x)^(1/3) * (1 + A(x)) )^(3/2).

1, 3, 9, 31, 117, 468, 1949, 8361, 36693, 163956, 743388, 3411576, 15816609, 73967637, 348517539, 1652896367, 7884305829, 37800279504, 182055056428, 880410972156, 4273376488956, 20811901707192, 101666716335912, 498035242836144, 2446003588237193, 12041562653655453

Offset: 0

Seiichi Manyama, Apr 04 2024

nonn

Cf. A106228, A262441, A370891.

Cf. A371725.

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

G.f.: B(x)^3 where B(x) is the g.f. of A106228.

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+3*k/2+3/2,n)/(n/3+k+1).

A053617 Number of permutations of length n which avoid the patterns 1234 and 1324.

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074, 176351644, 959658200, 5262988330, 29057961666, 161374413196, 900792925199, 5050924332096, 28434661250454, 160644331001476, 910455895039056, 5174722258676440, 29486753617569684

Offset: 0

Moa Apagodu, Mar 20 2000

nonn

These permutations have an "enumeration scheme" of depth 4, see D. Zeilberger's article in the links.
G.f. conjectured to be non-D-finite (see Albert et al. link). - Jay Pantone, Oct 01 2015
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 2>4, 3>4} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is the smallest. - Sergey Kitaev, Dec 10 2020

Andrew Baxter and Jay Pantone, Table of n, a(n) for n = 0..600 (terms n=1..100 from Andrew Baxter)
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Kremer, Darla and Shiu, Wai Chee, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Wikipedia, Permutation classes avoiding two patterns of length 4.
D. Zeilberger, Enumeration schemes and more importantly their automatic generation, Annals of Combinatorics 2 (1998) 185-195. The link is to an overview on Doron Zeilberger's home page; there is a local copy here [Pdf file only, no active links]

Cf. A032351, A053614, A106228, A165542, A165545, A257561, A257562.

More terms from Andrew Baxter, May 20 2011

A365243 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^3A(x)^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 22, 45, 99, 226, 515, 1168, 2670, 6186, 14467, 33985, 80105, 189636, 451060, 1077225, 2580979, 6201602, 14942480, 36098349, 87417956, 212159347, 515937882, 1257048536, 3068146679, 7500995555, 18366760161, 45037590888, 110588510089

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A101785, A106228, A112805.

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

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(n-k+1,n-3*k)/(n-k+1).

A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 3, 14, 77, 464, 2964, 19717, 135131, 947549, 6765642, 49022225, 359545750, 2664127354, 19913283809, 149968276974, 1136856855549, 8668000962927, 66428474900907, 511414514214628, 3953420853213504, 30674783555852576, 238808419235022293, 1864869207177530320

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A106228, A369479.
Cf. A143927, A369478.
Cf. A369440.

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

a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+2,k) * binomial(3*n-k+3,n-2*k).

A369479 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^3) ).

Original entry on oeis.org

1, 4, 25, 185, 1503, 12958, 116410, 1077872, 10213954, 98574454, 965545161, 9574235477, 95920415338, 969467658540, 9872949735243, 101211280459929, 1043597450013094, 10816134194658976, 112617367970103163, 1177413807406659659, 12355753915291229596

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Cf. A106228, A369477.

Cf. A365128, A369480.

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

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(4*n-k+4,n-2*k).

A108443 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k triple descents (i.e., ddd's).

Original entry on oeis.org

1, 2, 6, 3, 1, 21, 24, 15, 5, 1, 80, 150, 145, 84, 31, 7, 1, 322, 857, 1145, 949, 528, 202, 53, 9, 1, 1347, 4692, 8096, 8801, 6598, 3551, 1394, 398, 81, 11, 1, 5798, 25102, 53457, 72338, 68594, 47805, 25092, 10019, 3040, 692, 115, 13, 1, 25512, 132484, 337132

Offset: 0

Emeric Deutsch and Paul D. Hanna, Jun 10 2005

Row n has 2n-1 terms (n >= 1). Row sums yield A027307. Column 0 yields A106228.

			T(2,1) = 3 because we have uUddd, Uuddd and UdUddd.
Triangle begins:
    1;
    2;
    6,    3,    1;
   21,   24,   15,    5,    1;
   80,  150,  145,   84,   31,    7,    1;
  322,  857, 1145,  949,  528,  202,   53,    9,    1;

Alois P. Heinz, Rows n = 0..100, flattened
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A106228.

b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y-1, min(2, t+1))*`if`(t=2, z, 1)+
       b(x-1, y+2, 0)+b(x-2, y+1, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(3*n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Oct 06 2015

b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, Min[2, t + 1]]*If[t == 2, z, 1] + b[x - 1, y + 2, 0] + b[x - 2, y + 1, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[3*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

{T(n,k)=local(G=1+z*O(z^n)+t*O(t^k));for(k=1,n, G=1+z*(t+z-t*z)^2*G^3+z*(2-t)*(t+z-t*z)*G^2+2*z*(1-t)*G); polcoeff(polcoeff(G,n,z),k,t)}

G.f. G = G(t,z) satisfies G = 1 + z(t + z - tz)^2*G^3 + z(2-t)(t + z - tz)G^2 + 2z(1-t)G.

A215067 Number of Motzkin n-paths avoiding odd-numbered steps that are up steps.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 21, 37, 80, 146, 322, 602, 1347, 2563, 5798, 11181, 25512, 49720, 114236, 224540, 518848, 1027038, 2384538, 4748042, 11068567, 22150519, 51817118, 104146733, 244370806, 493012682, 1159883685, 2347796965, 5536508864, 11239697816, 26560581688, 54061835288

Offset: 0

David Scambler, Aug 02 2012

nonn

This sequence interleaves the counts of the closely related sequences A109081 and A106228.

a(n) is the number of (peakless) Motzkin paths of length n where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on n vertices where only vertices of the same parity can be matched. - Alexander Burstein, May 17 2021

			a(5) = 6: fUfFd, fUfDf, fUdUd, fUdFf, fFfUd, fFfFf showing odd-numbered steps in lower case.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A109081, A106228, A114465, A084075.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y) +b(x-1, y+1) +
      `if`(irem(x, 2)=1, 0, b(x-1, y-1)) ))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 04 2013

f[n_,x_,y_]:=f[n,x,y] = If[x>n||y<0,0,If[x==n&&y==0,1, If[EvenQ[x],0,f[n,x+1,y+1]] +f[n,x+1,y-1] + f[n,x+1,y]]]; Table[f[n,0,0],{n,0,35}]

{a(n)=polcoeff((1/x)*serreverse(x*(3+2*x+x^2 - sqrt((1+x^2)*(1+4*x+x^2)+x^2*O(x^n)))/(2*(1+x+x^2+x^2*O(x^n)))),n)} \\ Paul D. Hanna, Aug 02 2012

from mpmath import mp
mp.dps = 25; mp.pretty = True
def A215067(n) :
    m = n%2; r = n//2 if n>0 else 1
    return r^(1-m)*mp.hyper([-r,1-r-2*m,1+r+m],[(3-m)/2,(4-m)/2],1/4)
[int(A215067(i)) for i in (0..32)]  # Peter Luschny, Aug 03 2012

a(2*n) = Sum_{k=0..n} binomial(n+k-1,n-k) * binomial(n,k)/(n-k+1);
a(2*n+1) = Sum_{k=0..n} binomial(n+k+1,n-k) * binomial(n,k)/(n-k+1).
G.f.: (1/x)*Series_Reversion( x*(3+2*x+x^2 - sqrt((1+x^2)*(1+4*x+x^2)))/(2*(1+x+x^2)) ). - Paul D. Hanna, Aug 02 2012
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = (3+2*x+x^2 + sqrt((1+x^2)*(1+4*x+x^2)))/4. - Paul D. Hanna, Aug 02 2012
G.f. satisfies: Series_Reversion(x*A(x)) = x - x^2*F(-x) where F(x) = g.f. of A114465. - Paul D. Hanna, Aug 02 2012
a(n) = 3_F_2([-r,1-r-2*m,1+r+m],[(3-m)/2,(4-m)/2],1/4)*r^(1-m) for n>0 where m = n mod 2 and r = floor(n/2). - Peter Luschny, Aug 03 2012

A364723 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^4).

Original entry on oeis.org

1, 1, 2, 8, 38, 196, 1073, 6120, 35968, 216304, 1324676, 8232981, 51796538, 329229344, 2111031444, 13638557196, 88695018723, 580153216512, 3814285704000, 25192499164320, 167075960048996, 1112162062296061, 7428213584196010, 49766086788057256

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A000108, A106228, A300048, A364734.

Cf. A364739.

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

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+3*k,n-1-k) for n > 0.

A364734 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^5).

Original entry on oeis.org

1, 1, 2, 9, 48, 276, 1687, 10750, 70597, 474478, 3247844, 22563904, 158693152, 1127661358, 8083795761, 58390722901, 424562043703, 3104994695198, 22825260066996, 168564068029385, 1249985066423749, 9303815610715531, 69483859839881494, 520527161650519576

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A000108, A106228, A300048, A364723.

Cf. A364740.

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

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

A370891 G.f. satisfies A(x) = ( 1 + x * A(x)^(1/4) * (1 + A(x)^(3/4)) )^2.

Original entry on oeis.org

1, 4, 14, 52, 205, 844, 3588, 15632, 69434, 313264, 1431650, 6613732, 30834548, 144895284, 685566370, 3263309844, 15616322995, 75085908112, 362563417968, 1757412095456, 8548129677400, 41710100368160, 204110896990686, 1001480947876276, 4925833177966164

Offset: 0

Seiichi Manyama, Apr 04 2024

nonn

Cf. A000292, A371723.

Cf. A106228, A262441, A371724.

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

G.f.: B(x)^4 where B(x) is the g.f. of A106228.

a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(n/2+3*k/2+2,n)/(n/2+3*k/2+2).

cp's OEIS Frontend

A371724 G.f. satisfies A(x) = ( 1 + x * A(x)^(1/3) * (1 + A(x)) )^(3/2).

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A053617 Number of permutations of length n which avoid the patterns 1234 and 1324.

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A365243 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^3*A(x)^2).

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A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

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A369479 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^3) ).

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A108443 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k triple descents (i.e., ddd's).

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A215067 Number of Motzkin n-paths avoiding odd-numbered steps that are up steps.

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A364723 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x*A(x)^4).

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A364734 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x*A(x)^5).

A365243 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^3A(x)^2).

A364723 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^4).

A364734 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^5).