cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A120730 Another version of Catalan triangle A009766.

Original entry on oeis.org

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

Views

Author

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

Keywords

Comments

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

Examples

			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;
  ...

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • SageMath
    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

Formula

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

Views

Author

Manuel Kauers, Nov 18 2008

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • SageMath
    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

Formula

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) = 6*a(2n+1), a(2n+1) = 6*a(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

Views

Author

Philippe Deléham, Feb 05 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

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

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.

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

Views

Author

Philippe Deléham, Feb 08 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Maple
    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
  • Mathematica
    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 *)
  • SageMath
    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

Formula

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(2*n+2) = 8*a(2*n+1), a(2*n+1) = 8*a(2*n) - 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

Views

Author

Philippe Deléham, Feb 08 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • SageMath
    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

Formula

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) = 9*a(2n+1), a(2n+1) = 9*a(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

Views

Author

Philippe Deléham, Feb 10 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • SageMath
    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

Formula

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

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.

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

Views

Author

Philippe Deléham, Feb 10 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    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

Formula

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

A151253 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), (0, 0, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0)}.

Original entry on oeis.org

1, 4, 19, 91, 445, 2188, 10819, 53644, 266581, 1326646, 6609307, 32953033, 164397313, 820521562, 4096733707, 20459928259, 102203137741, 510621146326, 2551485015379, 12750737780587, 63725988599425, 318514790389294, 1592093707211299, 7958459733327676, 39783873348471745, 198883941062337328
Offset: 0

Views

Author

Manuel Kauers, Nov 18 2008

Keywords

Comments

Binomial transform of A151162. - Philippe Deléham, Feb 03 2009

Links

  • A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

Programs

  • Mathematica
    aux[i_Integer, j_Integer, k_Integer, n_Integer] := 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, k, -1 + n] + aux[-1 + i, j, -1 + k, -1 + n] + aux[-1 + i, j, k, -1 + n] + aux[i, j, -1 + 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, 10}]
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.