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 1-3 of 3 results.

A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   2;
  1, 3,  5,   6,   4;
  1, 4,  9,  14,  15,  10;
  1, 5, 14,  27,  38,  39,   25;
  1, 6, 20,  46,  79, 104,  102,   64;
  1, 7, 27,  72, 145, 229,  285,  270,  166;
  1, 8, 35, 106, 244, 446,  659,  784,  721,  436;
  1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;

Links

Crossrefs

Cf. A026269, A026270, A026288, A026289, A026290, A026291, A026292, A026293.
Cf. A026294, A026295, A026296, A026297, A026299, A026519, A026536, A026552.
Cf. A026584, A027926, A212342.

Programs

  • Magma
    f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n,k) // T = A026268
  if k eq 0 or n lt 3 then return 1;
  elif k eq 1 then return n-1;
  elif k eq 2 then return f(n);
  elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
  else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)
  • SageMath
    def T(n,k): # T = A026268
    if n<3 or k==0: return 1
    elif k==1: return n-1
    elif k==2: return (n^2 -n -2)//2 + int(n==2)
    elif k==n: return T(n-1, n-2) + T(n-1, n-1)
    else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022

Formula

From G. C. Greubel, Sep 24 2022: (Start)
T(n, 1) = A000027(n-1), n >= 1.
T(n, 2) = A212342(n-1), n >= 2.
T(n, n-1) = A026270(n), n >= 2.
T(n, n-2) = A026288(n), n >= 2.
T(n, n-3) = A026289(n), n >= 3.
T(n, n-4) = A026290(n), n >= 4.
T(n, n) = A026269(n), n >= 2.
T(n, floor(n/2)) = A026297(n), n >= 0.
T(2*n, n) = A026292(n).
T(2*n, n-1) = A026295(n), n >= 1.
T(2*n, n+1) = A026296(n), n >= 1.
T(2*n-1, n-1) = A026291(n), n >= 2.
T(3*n, n) = A026293(n), n >= 0.
T(4*n, n) = A026294(n), n >= 0.
Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)

Extensions

Updated by Clark Kimberling, Aug 29 2014
Indices of b-file corrected by Sidney Cadot, Jan 06 2023.

A102071 Pairwise sums of general ballot numbers (A002026).

Original entry on oeis.org

1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496, 35142703099692, 101055137177563
Offset: 1

Views

Author

Ralf Stephan, Dec 30 2004

Keywords

Links

Crossrefs

First differences of A005554. Partial sums of A026269. 3rd column of A348840.
Cf. A000108, A001006, A002057.

Programs

  • Mathematica
    CoefficientList[Series[(4x(1+x))/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,40}],x] (* Harvey P. Dale, Feb 26 2013 *)
  • Maxima
    a(n):=1/n*sum((binomial(j,n-1-j)+4*binomial(j,n-2-j)+3*binomial(j,n-3-j))*binomial(n,j),j,0,n); /* Vladimir Kruchinin, Mar 08 2016 */
  • PARI
    z='z+O('z^66); Vec(4*z*(1+z)/(1-z+sqrt(1-2*z-3*z^2))^2) \\ Joerg Arndt, Mar 08 2016

Formula

G.f.: (4*x*(1+x))/(1-x+sqrt(1-2*x-3*x^2))^2.
a(n) = (1/n) * Sum_{j=0..n} ((binomial(j,n-1-j)+4*binomial(j,n-2-j) + 3*binomial(j,n-3-j))*binomial(n,j)). - Vladimir Kruchinin, Mar 08 2016
a(n) ~ 4*3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 08 2016
a(n) = A001006(n+1) - A001006(n-1). - Gennady Eremin, Sep 23 2021
D-finite with recurrence (n+3)*a(n) + (-3*n-5)*a(n-1) + (-n+3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 01 2021
From Peter Bala, Feb 02 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A002057(k).
G.f.: x/(1 + x)*c(x/(1 + x))^4, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A026270 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-1), where T is the array in A026268.

Original entry on oeis.org

1, 2, 6, 15, 39, 102, 270, 721, 1941, 5262, 14354, 39372, 108528, 300482, 835278, 2330334, 6522882, 18313542, 51559506, 145530291, 411738723, 1167450066, 3316925794, 9441771081, 26923831029, 76901809810, 219992462862, 630245628681, 1808029517585
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A026269. Pairwise sums of A026122.

Formula

G.f.: -1 + 4z^2(1-z)(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.
Conjecture: (n+3)*a(n) +3*(-n-1)*a(n-1) +(-n-1)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 23 2013
Showing 1-3 of 3 results.

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