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 1-4 of 4 results.

A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793, 1, 17, 143, 795, 3285, 10717, 28435, 61463, 103049, 103049
Offset: 0

Views

Author

Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008

Keywords

Examples

			First few rows of the triangle:
  1
  1,  1
  1,  3,  3
  1,  5, 11,  11
  1,  7, 23,  45,  45
  1,  9, 39, 107, 197, 197
  1, 11, 59, 205, 509, 903, 903

Links

Crossrefs

Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.
Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.
Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).
Cf. A010683 (row sums), A186826 (rows reversed).
Cf. A239204, A247623.

Programs

  • Haskell
    a144944 n k = a144944_tabl !! n !! k
a144944_row n = a144944_tabl !! n
a144944_tabl = iterate f [1] where
   f us = vs ++ [last vs] where
     vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
-- Reinhard Zumkeller, May 11 2013
  • Mathematica
    t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q-1] + t[p-1, q] + t[p-1, q-1]; Flatten[Table[ t[p, q], {p,0,6}, {q,0, p}]] (* Jean-François Alcover, Dec 19 2011 *)
  • SageMath
    @CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kG. C. Greubel, Mar 11 2023

Formula

From G. C. Greubel, Mar 11 2023: (Start)
Sum_{k=0..n} T(n, k) = A010683(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)

A247629 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 4, 0, 5, 0, 1, 0, 12, 0, 7, 0, 1, 16, 0, 24, 0, 9, 0, 1, 0, 52, 0, 40, 0, 11, 0, 1, 68, 0, 116, 0, 60, 0, 13, 0, 1, 0, 236, 0, 216, 0, 84, 0, 15, 0, 1, 304, 0, 568, 0, 360, 0, 112, 0, 17, 0, 1, 0, 1108, 0, 1144, 0, 556, 0, 144
Offset: 0

Views

Author

Clark Kimberling, Sep 21 2014

Keywords

Examples

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
4 ... 0 ... 5 ... 0 ... 1
0 ... 12 .. 0 ... 7 ... 0 ...1
16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
T(4,2) counts these 5 paths given as vector sums applied to (0,0):
(1,1) + (1,1) + (1,1) + (1,-1)
(1,1) + (1,1) + (2,0)
(1,1) + (1,1) + (1,-1) + (1,1)
(1,1) + (2,0) + (1,1)
(1,1) + (1,-1) + (1,1) + (1,-1)

Links

Crossrefs

Cf. A247623, A247629, A026300, A006319 (1st column of this triangle).

Programs

  • Mathematica
    t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1,    t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
v = Flatten[u] (* A247629 sequence *)
Map[Total, u] (* A247630 *)

A247630 Number of paths from (0,0) to the line x = n, each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 50, 104, 258, 552, 1362, 2972, 7306, 16172, 39650, 88720, 217090, 489872, 1196834, 2719028, 6634890, 15157188, 36949266, 84799992, 206549250, 475894200, 1158337650, 2677788492, 6513914634, 15102309468, 36718533570, 85347160608
Offset: 0

Views

Author

Clark Kimberling, Sep 21 2014

Keywords

Comments

a(n) = sum of the numbers in row n of the triangle at A247629.

Examples

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
4 ... 0 ... 5 ... 0 ... 1
0 ... 12 .. 0 ... 7 ... 0 ...1
16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
T(4,2) counts these 5 paths given as vector sums applied to (0,0):
(1,1) + (1,1) + (1,1) + (1,-1)
(1,1) + (1,1) + (2,0)
(1,1) + (1,1) + (1,-1) + (1,1)
(1,1) + (2,0) + (1,1)
(1,1) + (1,-1) + (1,1) + (1,-1)
a(4) = 4 + 0 + 5 + 0 + 1 = 10.

Links

Crossrefs

Cf. A247629, A247623.

Programs

  • Mathematica
    t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1,    t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
v = Flatten[u] (* A247629 sequence *)
Map[Total, u] (* A247630 *)

Formula

Conjecture: -(n+1)*(n-2)*a(n) -(n-1)*(n-4)*a(n-1) +2*(3*n-2)*(n-2)*a(n-2) +2*(3*n-5)*(n-3)*a(n-3) +(-n^2+7*n-2)*a(n-4) -(n-1)*(n-6)*a(n-5)=0. - R. J. Mathar, Sep 23 2014

A247622 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 5, 0, 1, 0, 11, 0, 7, 0, 1, 11, 0, 23, 0, 9, 0, 1, 0, 45, 0, 39, 0, 11, 0, 1, 45, 0, 107, 0, 59, 0, 13, 0, 1, 0, 197, 0, 205, 0, 83, 0, 15, 0, 1, 197, 0, 509, 0, 347, 0, 111, 0, 17, 0, 1, 0, 903, 0, 1061, 0, 541, 0, 143, 0
Offset: 0

Views

Author

Clark Kimberling, Sep 21 2014

Keywords

Examples

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
3 ... 0 ... 5 ... 0 ... 1
0 ... 11 .. 0 ... 7 ... 0 ...1
11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
T(3,1) counts these 3 paths given as vector sums applied to (0,0):
(1,1) + (1,-1), (2,0), (1,-1) + (1,1).

Links

Crossrefs

Cf. A247623, A247629, A026300, A001003 (1st column of this triangle).

Programs

  • Mathematica
    t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1, t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = t[n - 1, 1]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}];
v = Flatten[u] (* A247622 sequence *)
TableForm[u]   (* A247622 array *)
Map[Total, u]  (* A247623 *)
Showing 1-4 of 4 results.

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