A052709 -id:A052709 - cp's OEIS frontend

A223092 Triangle read by rows: let T(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,0), (1,-1) and (0,-1); n-th row of triangle gives T(n,n), T(n,n-1), ..., T(n,0).

1, 1, 2, 1, 4, 7, 1, 6, 18, 29, 1, 8, 33, 86, 133, 1, 10, 52, 179, 431, 650, 1, 12, 75, 316, 978, 2238, 3319, 1, 14, 102, 505, 1874, 5406, 11941, 17498, 1, 16, 133, 754, 3235, 11020, 30241, 65086, 94525, 1, 18, 168, 1071, 5193, 20202, 64698, 171045, 360897, 520508, 1, 20, 207, 1464, 7896, 34362, 124455, 380400, 977040, 2029490, 2910895

Offset: 0

N. J. A. Sloane, Mar 29 2013

			Triangle begins:
[1]
[1, 2]
[1, 4, 7]
[1, 6, 18, 29]
[1, 8, 33, 86, 133]
[1, 10, 52, 179, 431, 650]
[1, 12, 75, 316, 978, 2238, 3319]
...
The T(n,k) array begins:
4:  0  0  0  0   1  10 ...
3:  0  0  0  1   8  52 ...
2:  0  0  1  6  33 179 ...
1:  0  1  4 18  86 431 ...
0:  1  2  7 29 133 650 ...
-------------------------
k/n:0  1  2  3   4   5 ...
T(5,2) = T(5,3) + T(4,3) + T(4,2) + T(4,1) = 52 + 8 + 33 + 86 = 179.- _Philippe Deléham_, Mar 29 2013
This is also Dziemianczuk's array N(-i,i+j) read by antidiagonals:
1,2,7,29,133,650,3319,17498, ...
1,4,18,86,431,2238,11941,65086, ...
1,6,33,179,978,5406,30241,171045, ...
1,8,52,316,1874,11020,64698,380400, ...
1,10,75,505,3235,20202,124455,761160, ...
... - _N. J. A. Sloane_, Dec 05 2013

Alois P. Heinz, Rows n = 0..140, flattened
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013
N. J. A. Sloane, Illustration of initial terms of A223092 and A064641

Cf. A064641 (T(n,0)), A071943, A052709.

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
      `if`(n<0 or k<0 or k>n, 0, add(T(n-l[1], k-l[2]),
       l=[[1, 1], [1, 0], [1, -1], [0, -1]]) ))
    end:
seq(seq(T(n, n-j), j=0..n), n=0..10);  # Alois P. Heinz, Apr 08 2013

max = 10; T[0, 0] = 1; T[n_ /; n >= 0, k_ /; 0 <= k <= max] := T[n, k] = T[n, k+1]+T[n-1, k+1]+T[n-1, k]+T[n-1, k-1]; T[n_, k_] = 0; Table[Table[T[n, k], {k, n, 0, -1}], {n, 0, max}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

T(n,k) = T(n,k+1) + T(n-1,k+1) + T(n-1,k) + T(n-1,k-1). - Philippe Deléham, Mar 29 2013

A260772 Certain directed lattice paths.

Original entry on oeis.org

1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.

Cf. A122868, A260774, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

# A260772 satisfies a 4th-order recurrence that can be reduced
# to a 2nd-order recurrence given in this program t:
t := proc(n) options remember;
if n <= 1 then
    [-1/2, 0, 1, 4][2*n+2]
  else
    (16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1))
       /( n*(2*n+1)*(5*n-7) )
  fi
end:
A260772 := proc(n)
t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n)
end:
seq(A260772(i),i=0..100);
# Mark van Hoeij, Jul 14 2022

a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */

a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019

G.f.: P1(x) = (2*(1-x)/3)/x - ((2*sqrt(1-5*x-2*x^2)/3)/x)*sin((Pi/6 + arccos(((20*x^3-6*x^2+15*x-2)/2)/(1-5*x-2*x^2)^(3/2))/3)). - See Dziemianczuk (2014), Proposition 11.
a(n) = (1/n)*Sum_{j=0..(n+1)/4} (-1)^j*C(n,j)*C(3*n-4*j,n-4*j+1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
n*(n+1)*(25*n^2-70*n+21)*a(n) - 30*(7*n-15)*n*a(n-1) + (-1100*n^4+5280*n^3-6424*n^2-1188*n+3816)*a(n-2) + 120*(n+2)*(n-3)*a(n-3) - 16*(n-3)*(n-4)*(25*n^2-20*n-24)*a(n-4) = 0. - Mark van Hoeij, Jul 14 2022
a(n) ~ 2^(n - 1/2) * phi^((10*n - 1)/4) / (sqrt(Pi) * 5^(1/4) * sqrt(phi^(3/2) - 2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 15 2022

More terms from Lars Blomberg, Aug 01 2015

A364477 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^7.

Original entry on oeis.org

1, 1, 3, 14, 76, 448, 2791, 18078, 120516, 821435, 5698422, 40101623, 285583775, 2054272430, 14903954415, 108932920861, 801350333186, 5928653489398, 44084056075057, 329279673851792, 2469493161891742, 18588339309502760, 140383789476473354

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A001002, A001764, A006605, A052709, A143330, A364473.

Cf. A364476.

a(n) = sum(k=0, n\2, binomial(2*n+3*k, k)*binomial(2*n+2*k, n-2*k)/(n+4*k+1));

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

A369208 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2) ).

Original entry on oeis.org

1, 2, 8, 38, 200, 1122, 6576, 39790, 246672, 1558658, 10001592, 64997814, 426922392, 2829624514, 18901301984, 127115260894, 859978039840, 5848754717314, 39964745880552, 274231943135686, 1888891689752680, 13055393137141282, 90517646431869328

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A218045, A369263.
Cf. A052709, A109081.
Cf. A370242.

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

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

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

a(n) = (1/(n+1)) *[x^n] ( 1/(1-x)^2 * (1+x^2) )^(n+1). - Seiichi Manyama, Feb 14 2024

A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).

Original entry on oeis.org

1, 1, 4, 13, 53, 220, 968, 4373, 20271, 95705, 458904, 2228220, 10934524, 54143848, 270189008, 1357428997, 6860264323, 34853234867, 177900211204, 911867479717, 4691701977973, 24222505191984, 125448280976224, 651555603531308, 3392951906596708, 17711433386188300

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A052709, A369262.

Cf. A218045, A370243.

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

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

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

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A108916 Triangle of Schroeder paths counted by number of diagonal steps not preceded by an east step.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 9, 3, 1, 31, 36, 18, 4, 1, 113, 155, 90, 30, 5, 1, 431, 678, 465, 180, 45, 6, 1, 1697, 3017, 2373, 1085, 315, 63, 7, 1, 6847, 13576, 12068, 6328, 2170, 504, 84, 8, 1, 28161, 61623, 61092, 36204, 14238, 3906, 756, 108, 9, 1, 117631, 281610, 308115, 203640, 90510, 28476, 6510, 1080, 135, 10, 1

Offset: 0

David Callan, Jul 25 2005

T(n,k) = number of Schroeder (= underdiagonal Delannoy) paths of steps east(E), north(N) and diagonal (D) (i.e., northeast) from (0,0) to (n,n) containing k Ds not preceded by an E. Also, T(n,k) = number of Schroeder paths from (0,0) to (n,n) containing k Ds not preceded by an N. This is because there is a simple bijection on Schroeder paths that interchanges the statistics "# Ds not preceded by an E" and "# Ds not preceded by an N": for each E and its matching N, interchange the diagonal segments (possibly empty) immediately following them (a diagonal segment is a maximal sequence of contiguous Ds).

			Table begins:
\ k..0...1...2...3...4...
n\
0 |..1
1 |..1...1
2 |..3...2...1
3 |..9...9...3...1
4 |.31..36..18...4...1
5 |113.155..90..30...5...1
The paths ENDD, DEND, DDEN each have 2 Ds not preceded by an E and so T(3,2)=3.

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 is A052709 shifted left. Cf. A110446.

G[z_, t_] = (1-t*z - ((1-t*z)^2 + 4z(-1-z+t*z))^(1/2))/(2z(1+z-t*z));
CoefficientList[#, t]& /@ CoefficientList[G[z, t] + O[z]^11, z] // Flatten (* Jean-François Alcover, Oct 06 2019 *)

G.f.: G(z,t) = Sum_{n>=k>=0} T(n,k)*z^n*t^k satisfies G = 1 + z*t*G + z(1 + z - z*t)G^2 with solution G(z,t) = (1 - t*z - ((1 - t*z)^2 + 4*z*(-1 - z + t*z))^(1/2))/(2*z*(1 + z - t*z)).

A238578 Expansion of -(-4x^4 + sqrt(-4x^2-4x+1) (2x^3+x^2-2x) -12x^3-7x^2+2x) / (sqrt(-4x^2-4x+1) (4x^3+8x^2+3x-1) - 4x^3-8x^2-3x+1).

Original entry on oeis.org

0, 1, 3, 11, 45, 191, 833, 3695, 16593, 75199, 343233, 1575551, 7265921, 33637631, 156234497, 727681791, 3397475585, 15896054783, 74512968705, 349859309567, 1645121398785, 7746058698751, 36516283891713, 172332643868671, 814108326764545, 3849410342715391

Offset: 0

Vladimir Kruchinin, Mar 01 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A052709.

Table[Sum[Binomial[n - 1, k - 1] * Sum[Binomial[k, n - k - i] * Binomial[k + i - 1, k - 1], {i, 0, n - k}], {k, n}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 02 2014 *)
CoefficientList[Series[-(-4*x^4 + Sqrt[-4*x^2-4*x+1]*(2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x)/(Sqrt[-4*x^2-4*x+1]*(4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1), {x, 0, 50}], x] (* G. C. Greubel, Jun 01 2017 *)

a(n):= sum((sum(binomial(k,n-k-i)*binomial(k+i-1,k-1), i,0,n-k)) *binomial(n-1,k-1), k,1,n);

my(x='x+O('x^50)); concat([0], Vec(-(-4*x^4 + sqrt(-4*x^2-4*x+1)*(2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x)/(sqrt(-4*x^2-4*x+1)*(4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1))) \\ G. C. Greubel, Jun 01 2017

for(n=0,25, print1(sum(k=1,n, binomial(n-1,k-1)*sum(i=0,n-k, binomial(k,n-k-i)*binomial(k+i-1,k-1))), ", ")) \\ G. C. Greubel, Jun 01 2017

a(n) = Sum_{k=1..n} Sum_{i=0..(n-k)} C(k,n-k-i)*C(k+i-1,k-1)*C(n-1,k-1).
G.f.: A(x) = x*(F(x)-x)*F'(x)/F(x)^2, where F(x) = (1-sqrt(-4*x^2-4*x+1))/(2*x+2), F(x) is the g.f. of A052709.
D-finite with recurrence: (for n>5): (n-5)*(n-1)*a(n) = (3*n^2 - 20*n + 23)*a(n-1) + 2*(n-2)*(4*n-19)*a(n-2) + 4*(n-4)*(n-3)*a(n-3). - Vaclav Kotesovec, Mar 03 2014
a(n) ~ (2 + 2*sqrt(2))^n / (2^(5/4) * sqrt(1+sqrt(2)) * sqrt(Pi*n)). - Vaclav Kotesovec, Mar 03 2014

A364473 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^6.

Original entry on oeis.org

1, 1, 3, 13, 65, 353, 2024, 12057, 73890, 462851, 2950261, 19073921, 124776881, 824409052, 5493384031, 36874564529, 249114808794, 1692489908494, 11556616157589, 79265016880139, 545860966841247, 3772800724433931, 26162662010039826, 181974370638420829

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A001002, A001764, A006605, A052709, A143330, A364477.

Cf. A364472, A364474.

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

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

A366025 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x^5) ).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 139, 465, 1595, 5577, 19804, 71228, 258946, 950030, 3513050, 13079920, 48993149, 184490361, 698020080, 2652192675, 10115878915, 38717526745, 148655862210, 572412768275, 2209969761924, 8553073927858, 33176952295730, 128960722306128

Offset: 0

Seiichi Manyama, Sep 26 2023

nonn

Cf. A006318, A052709, A071969, A365268.

Cf. A365760, A366024.

CoefficientList[InverseSeries[Series[x(1-x)/(1+x^5),{x,0,28}],x]/x,x] (* Stefano Spezia, Sep 26 2023 *)

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*n-5*k+1, n-4*k)/(2*n-5*k+1));

Vec(serreverse(x*(1-x)/(1+x^5)+O(x^30))/x) \\ Michel Marcus, Sep 26 2023

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

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(2*n-5*k+1,n-4*k)/(2*n-5*k+1) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+1,k) * binomial(2*n-5*k,n-5*k).

A039977 An example of a d-perfect sequence.

Original entry on oeis.org

1, 2, 0, 0, 1, 1, 2, 1, 1, 0, 1, 2, 2, 0, 0, 1, 0, 2, 2, 1, 0, 0, 2, 2, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 0, 0, 2, 0, 1, 1, 2, 2, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 2, 1, 2, 2, 0, 2, 1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 0, 1, 1, 2, 1, 1, 0, 1, 2, 2, 0, 0, 1, 0, 0, 2, 0, 1, 1, 2, 2, 0, 2, 1, 1, 0, 2, 2, 1, 1

Offset: 1

N. J. A. Sloane

nonn

D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

a(n) = ((-1)^(n+1)*A052709(n)) mod 3. - Christian G. Bower, Jun 12 2005

More terms from Christian G. Bower, Jun 12 2005

cp's OEIS Frontend

A223092 Triangle read by rows: let T(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,0), (1,-1) and (0,-1); n-th row of triangle gives T(n,n), T(n,n-1), ..., T(n,0).

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A260772 Certain directed lattice paths.

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A364477 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^7.

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PARI

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A369208 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2) ).

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PARI

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A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).

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PARI

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A108916 Triangle of Schroeder paths counted by number of diagonal steps not preceded by an east step.

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Mathematica

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A238578 Expansion of -(-4*x^4 + sqrt(-4*x^2-4*x+1) * (2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x) / (sqrt(-4*x^2-4*x+1) * (4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1).

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Maxima

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PARI

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A364473 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^6.

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A364477 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^7.

A238578 Expansion of -(-4x^4 + sqrt(-4x^2-4x+1) (2x^3+x^2-2x) -12x^3-7x^2+2x) / (sqrt(-4x^2-4x+1) (4x^3+8x^2+3x-1) - 4x^3-8x^2-3x+1).

A364473 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^6.