A151281 -id:A151281 - cp's OEIS frontend

A120730 Another version of Catalan triangle A009766.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.
Aerated version gives A165408. - Philippe Deléham, Sep 22 2009
T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jun 19 2011
With zeros omitted: 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - Philippe Deléham, Nov 02 2011

			As a triangle, this begins:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  2,  1;
  0,  0,  2,  3,  1;
  0,  0,  0,  5,  4,  1;
  0,  0,  0,  5,  9,  5,  1;
  0,  0,  0,  0, 14, 14,  6,  1;
  ...

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

Cf. A000007, A000096, A000108, A001405, A001477, A003161, A005586, A005587.
Cf. A008313, A009766, A039598, A039599, A084938, A099376, A105523, A121725.
Cf. A126087, A128386, A121724, A128387, A129123, A132373, A132374, A132375.
Cf. A132889, A151162, A151254, A151281, A156195, A156361, A156362, A156566.
Cf. A156577, A165408, A357654.

A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
[A120730(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Nov 07 2022

G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form  # Emeric Deutsch, Jun 19 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, 2*k-n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Oct 13 2022

b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
T[n_, k_] := b[n, 2 k - n];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 21 2022, after Alois P. Heinz *)
T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 07 2022 *)

def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 07 2022

G.f.: G(t,z) = 4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2))). - Emeric Deutsch, Jun 19 2011
Sum_{k=0..n} x^k*T(n,n-k) = A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively. [corrected by Philippe Deléham, Oct 16 2008]
T(2*n,n) = A000108(n); A000108: Catalan numbers.
From Philippe Deléham, Oct 18 2008: (Start)
Sum_{k=0..n} T(n,k)^2 = A000108(n) and Sum_{n>=k} T(n,k) = A000108(k+1).
Sum_{k=0..n} T(n,k)^3 = A003161(n).
Sum_{k=0..n} T(n,k)^4 = A129123(n). (End)
Sum_{k=0..n}, T(n,k)*x^k = A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 10 2009
From G. C. Greubel, Nov 07 2022: (Start)
T(n, k) = 0 if n > 2*k, otherwise binomial(n, k)*(2*k-n+1)/(k+1).
Sum_{k=0..n} (-1)^k*T(n,k) = A105523(n).
Sum_{k=0..n} (-1)^k*T(n,k)^2 = -A132889(n), n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A357654(n).
T(n, n-1) = A001477(n).
T(n, n-2) = [n=2] + A000096(n-3), n >= 2.
T(n, n-3) = 2*[n<5] + A005586(n-5), n >= 3.
T(n, n-4) = 5*[n<7] - 2*[n=4] + A005587(n-7), n >= 4.
T(2*n+1, n+1) = A000108(n+1), n >= 0.
T(2*n-1, n+1) = A099376(n-1), n >= 1. (End)

A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

Original entry on oeis.org

1, 4, 20, 96, 480, 2368, 11840, 58880, 294400, 1468416, 7342080, 36667392, 183336960, 916144128, 4580720640, 22896574464, 114482872320, 572320645120, 2861603225600, 14306741583872, 71533707919360, 357650927714304, 1788254638571520, 8941026626502656, 44705133132513280, 223522175800311808

Offset: 0

Manuel Kauers, Nov 18 2008

Hankel transform is 4^binomial(n+1,2). - Philippe Deléham, Feb 01 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2009.

Cf. A000108, A001405, A120730, A151162, A151281, A151403, A156058.

[n le 3 select Factorial(n+2)/6 else (5*n*Self(n-1) + 16*(n-3)*Self(n-2) - 80*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1+i, -1+j, -1+k, -1+n] + aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1 + n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n, 0, 30}]
a[n_]:= a[n]= If[n<3, (n+3)!/3!, (5*(n+1)*a[n-1] +16*(n-2)*a[n-2] -80*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A151254
    if (n==0): return 1
    elif (n%2==1): return 5*a(n-1) - 4^((n-1)/2)*catalan_number((n-1)/2)
    else: return 5*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k)*4^k. - Philippe Deléham, Feb 01 2009
From Philippe Deléham, Feb 02 2009: (Start)
a(2n+2) = 5*a(2n+1), a(2n+1) = 5*a(2n) - 4^n*A000108(n) = 5*a(2n) - A151403(n).
G.f.: (sqrt(1-16*x^2) + 8*x - 1)/(8*x*(1-5*x)). (End)
a(n) = (5*(n+1)*a(n-1) + 16*(n-2)*a(n-2) - 80*(n-2)*a(n-3))/(n+1). - G. C. Greubel, Nov 09 2022

A156195 a(2n+2) = 6a(2n+1), a(2n+1) = 6a(2n) - 5^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 5, 30, 175, 1050, 6250, 37500, 224375, 1346250, 8068750, 48412500, 290343750, 1742062500, 10450312500, 62701875000, 376177734375, 2257066406250, 13541839843750, 81251039062500, 487496738281250, 2924980429687500, 17549718554687500, 105298311328125000

Offset: 0

Philippe Deléham, Feb 05 2009

nonn

Hankel transform is 5^C(n+1,2). - Philippe Deléham, Feb 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156058.

[n le 3 select Factorial(n+3)/24 else (6*n*Self(n-1) + 20*(n-3)*Self(n-2) - 120*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

A156195 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        6*procname(n-1);
    else
        nh := floor(n/2) ;
        6*procname(n-1)-5^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

CoefficientList[Series[(Sqrt[1-20x^2]+10x-1)/(10x(1-6x)),{x,0,30}],x] (* Harvey P. Dale, Oct 21 2016 *)

def a(n): # a = A156195
    if (n==0): return 1
    elif (n%2==1): return 6*a(n-1) - 5^((n-1)/2)*catalan_number((n-1)/2)
    else: return 6*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k)*5^k.
G.f.: (sqrt(1-20*x^2) +10*x -1)/(10*x*(1-6*x)). - Philippe Deléham, Feb 05 2009
(n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016

Corrected and extended by Harvey P. Dale, Oct 21 2016

A156361 a(2n+2) = 7a(2n+1), a(2n+1) = 7a(2n) - 6^n*A000108(n), a(0) = 1.

Original entry on oeis.org

1, 6, 42, 288, 2016, 14040, 98280, 686880, 4808160, 33638976, 235472832, 1647983232, 11535882624, 80745019776, 565215138432, 3956385876480, 27694701135360, 193860506096640, 1357023542676480, 9499115800977408

Offset: 0

Philippe Deléham, Feb 08 2009

nonn

Hankel transform is 6^C(n+1, 2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Cf. A000108, A001405, A156128, A151162, A156195, A151254, A151281.

[n le 3 select Factorial(n+4)/120 else (7*n*Self(n-1) + 24*(n-3)*Self(n-2) - 168*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

A156361 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        7*procname(n-1);
    else
        nh := floor(n/2) ;
        7*procname(n-1)-6^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

a[n_]:= a[n]= If[n==0, 1, 7*a[n-1] -If[EvenQ[n], 0, 6^((n-1)/2)* CatalanNumber[(n-1)/2]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 04 2022 *)

def a(n): # a = A156361
    if (n==0): return 1
    elif (n%2==1): return 7*a(n-1) - 6^((n-1)/2)*catalan_number((n-1)/2)
    else: return 7*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum{k=0..n} A120730(n,k) * 6^k.

(n+1)*a(n) = 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 7, 56, 441, 3528, 28126, 225008, 1798349, 14386792, 115060722, 920485776, 7363180314, 58905442512, 471228010428, 3769824083424, 30158239367445, 241265914939560, 1930119075851050, 15440952606808400, 123527424655229966

Offset: 0

Philippe Deléham, Feb 08 2009

nonn

Hankel transform is 7^C(n+1,2).

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

Cf. A000108, A001405, A151162, A156195, A151254, A151281, A156266, A156361.

[n le 3 select Factorial(n+5)/720 else (8*n*Self(n-1) + 28*(n-3)*Self(n-2) - 224*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 8*a[n-1] -7^((n-1)/2)*CatalanNumber[(n-1)/2], 8*a[n-1]]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A156362
    if (n==0): return 1
    elif (n%2==1): return 8*a(n-1) - 7^((n-1)/2)*catalan_number((n-1)/2)
    else: return 8*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 7^k.

a(n) = ( 8*(n+1)*a(n-1) + 28*(n-2)*a(n-2) - 224*(n-2)*a(n-3) )/(n+1). - G. C. Greubel, Nov 09 2022

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 8, 72, 640, 5760, 51712, 465408, 4186112, 37675008, 339017728, 3051159552, 27459059712, 247131537408, 2224149233664, 20017343102976, 180155188248576, 1621396694237184, 14592546256715776, 131332916310441984

Offset: 0

Philippe Deléham, Feb 10 2009

nonn

Hankel transform is 8^C(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001405, A151162, A151254, A151281.
Cf. A156195, A156270, A156361, A156362, A156577.
Cf. A001018, A120730.

a[0] = 1; a[1] = 8; a[2] = 72; a[n_] := a[n] = (-288*(n-2)*a[n-3] + 32*(n-2)*a[n-2] + 9*(n+1)*a[n-1])/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Nov 15 2016 *)
a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 9*a[n-1] - 8^((n-1)/2)*CatalanNumber[(n- 1)/2], 9*a[n-1]]]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 18 2022 *)

def a(n): # a = A156566
    if (n==0): return 1
    elif (n%2==1): return 9*a(n-1) - 8^((n-1)/2)*catalan_number((n-1)/2)
    else: return 9*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, May 18 2022

a(n) = Sum_{k=0..n} A120730(n,k)*8^k.

A156577 a(2n+2) = 10a(2n+1), a(2n+1) = 10a(2n) - 9^n*A000108(n), a(0) = 1.

Original entry on oeis.org

1, 9, 90, 891, 8910, 88938, 889380, 8890155, 88901550, 888923646, 8889236460, 88889884542, 888898845420, 8888918303988, 88889183039880, 888889778505099, 8888897785050990, 88888916293698870, 888889162936988700

Offset: 0

Philippe Deléham, Feb 10 2009

nonn

Hankel transform is 9^binomial(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156195, A156273, A156361, A156362, A156566.

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 10*a[n-1] -9^((n-1)/2)*CatalanNumber[(n-1)/2], 10*a[n-1] ]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 04 2022 *)

def a(n): # a = A156577
    if (n==0): return 1
    elif (n%2==1): return 10*a(n-1) - 9^((n-1)/2)*catalan_number((n-1)/2)
    else: return 10*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Jan 04 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 9^k.

A368164 Number of nondeterministic Dyck bridges of length 2*n.

Original entry on oeis.org

1, 7, 63, 583, 5407, 50007, 460815, 4231815, 38745279, 353832631, 3224323183, 29328492519, 266364307231, 2416023142423, 21890268365007, 198151683934023, 1792260214473087, 16199857938091383, 146342491104098607, 1321339563995562663, 11925412051760977887, 107590261672922633943

Offset: 0

Michael Wallner, Dec 14 2023

nonn

In nondeterministic walks (N-walks) the steps are sets and called N-steps. N-walks start at 0 and are concatenations of such N-steps such that all possible extensions are explored in parallel. The nondeterministic Dyck step set is { {-1}, {1}, {-1,1} }. Such an N-walk is called an N-bridge if it contains at least one trajectory that is a classical bridge, i.e., starts and ends at 0 (for more details see the de Panafieu-Wallner article).

			The a(1)=7 N-bridges of length 2 are
           /              /    /
/\,    ,  /\,    ,  /\,  / ,  /\
     \/        \/   \    \/   \/
                \    \         \

Élie de Panafieu and Michael Wallner, Combinatorics of nondeterministic walks, arXiv:2311.03234 [math.CO], 2023.

Cf. A151281 (nondeterministic Dyck meanders), A368234 (nondeterministic Dyck excursions), A000244 (nondeterministic Dyck walks).

G.f.: (1-6*t)/(sqrt(1-8*t)*(1-9*t)).
From Joseph M. Shunia, May 09 2024: (Start)
a(n) = A089022(n) + Sum_{k=0..n-1} binomial(2*n, k)*2^(2*n-k).
a(n) = A000244(2*n) - Sum_{k=n+1..2*n} binomial(2*n, k)*2^(2*n-k+1). (End)

A368234 Number of nondeterministic Dyck excursions of length 2*n.

Original entry on oeis.org

1, 4, 28, 224, 1888, 16320, 143040, 1264128, 11230720, 100124672, 894785536, 8010072064, 71794294784, 644079468544, 5782109208576, 51934915067904, 466666751655936, 4194593964294144, 37711993926844416, 339119962067042304, 3049961818869989376, 27434013235435536384

Offset: 0

Michael Wallner, Dec 18 2023

nonn

In nondeterministic walks (N-walks) the steps are sets and called N-steps. N-walks start at 0 and are concatenations of such N-steps such that all possible extensions are explored in parallel. The nondeterministic Dyck step set is { {-1}, {1}, {-1,1} }. Such an N-walk is called an N-excursion if it contains at least one trajectory that is a classical excursion, i.e., never crosses the x-axis, and starts and ends at 0 (for more details see the de Panafieu-Wallner article).

			The a(1)=4 N-bridges of length 2 are
      /         /
/\,  /\,  /\,  /\
          \    \/
           \    \

Élie de Panafieu and Michael Wallner, Combinatorics of nondeterministic walks, arXiv:2311.03234 [math.CO], 2023.

Cf. A151281 (Nondeterministic Dyck meanders), A368164 (Nondeterministic Dyck bridges), A000244 (Nondeterministic Dyck walks).

G.f.: (1-8*x-(1-12*x)*sqrt(1-8*x))/(8*x*(1-9*x)).

cp's OEIS Frontend

A120730 Another version of Catalan triangle A009766.

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A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

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A156195 a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1.

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A156361 a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.

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A156362 a(2*n+2) = 8*a(2*n+1), a(2*n+1) = 8*a(2*n) - 7^n*A000108(n), a(0)=1.

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A156566 a(2n+2) = 9*a(2n+1), a(2n+1) = 9*a(2n) - 8^n*A000108(n), a(0)=1.

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SageMath

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A156577 a(2*n+2) = 10*a(2*n+1), a(2*n+1) = 10*a(2*n) - 9^n*A000108(n), a(0) = 1.

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A156195 a(2n+2) = 6a(2n+1), a(2n+1) = 6a(2n) - 5^n*A000108(n), a(0)=1.

A156361 a(2n+2) = 7a(2n+1), a(2n+1) = 7a(2n) - 6^n*A000108(n), a(0) = 1.

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.

A156577 a(2n+2) = 10a(2n+1), a(2n+1) = 10a(2n) - 9^n*A000108(n), a(0) = 1.