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.

A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

Original entry on oeis.org

1, 3, 1, 6, 3, 2, 10, 6, 5, 3, 15, 10, 9, 7, 4, 21, 15, 14, 12, 9, 5, 28, 21, 20, 18, 15, 11, 6, 36, 28, 27, 25, 22, 18, 13, 7, 45, 36, 35, 33, 30, 26, 21, 15, 8, 55, 45, 44, 42, 39, 35, 30, 24, 17, 9, 66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10
Offset: 0

Views

Author

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

Keywords

Examples

			First few rows of the triangle =
   1;
   3,  1;
   6,  3,  2;
  10,  6,  5,  3;
  15, 10,  9,  7,  4;
  21, 15, 14, 12,  9,  5;
  28, 21, 10, 18, 15, 11,  6;
  36, 28, 27, 25, 22, 18, 13,  7;
  45, 36, 35, 33, 30, 26, 21, 15,  8;
  55, 45, 44, 42, 39, 35, 30, 24, 17,  9;
  66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
  78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
  91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
  ...

Links

Crossrefs

Cf. A000012, A000096, A000217, A006527, A034828, A055998, A145677, A134479.

Programs

  • Mathematica
    T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2021 *)
  • Sage
    def A158822(n,k):
    if (k==0): return binomial(n+2, 2)
    else: return (n-k+1)*(n+k)/2
flatten([[A158822(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 26 2021

Formula

Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.
From G. C. Greubel, Dec 26 2021: (Start)
T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).
Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).
T(n, 1) = A000217(n).
T(n, 2) = A000096(n-1).
T(n, 3) = A055998(n-2).
T(2*n, n) = A134479(n). (End)

Extensions

Definition corrected by Michael Somos, Nov 05 2011

A158841 Triangle read by rows, matrix product of A145677 * A004736.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 13, 9, 6, 3, 21, 16, 12, 8, 4, 31, 25, 20, 15, 10, 5, 43, 36, 30, 24, 18, 12, 6, 57, 49, 42, 35, 28, 21, 14, 7, 73, 64, 56, 48, 40, 32, 24, 16, 8, 91, 81, 72, 63, 54, 45, 36, 27, 18, 9
Offset: 1

Views

Author

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

Keywords

Examples

			First few rows of the triangle:
    1;
    3,   1;
    7,   4,   2;
   13,   9,   6,   3;
   21,  16,  12,   8,   4;
   31,  25,  20,  15,  10,  5;
   43,  36,  30,  24,  18, 12,  6;
   57,  49,  42,  35,  28, 21, 14,  7;
   73,  64,  56,  48,  40, 32, 24, 16,  8;
   91,  81,  72,  63,  54, 45, 36, 27, 18,  9;
  111, 100,  90,  80,  70, 60, 50, 40, 30, 20, 10;
  133, 121, 110,  99,  88, 77, 66, 55, 44, 33, 22, 11;
  157, 144, 132, 120, 108, 96, 84, 72, 60, 48, 36, 24, 12;
  ...

Crossrefs

Cf. A145677, A002061 (column k=1), A158842 (row sums).

Programs

  • Maple
    A145677 := proc(n,k)
        if n <0 or k < 0 or k > n then
                0;
        elif k = 0 then
                1;
        elif k = n then
                n ;
        else
                0 ;
        end if;
end proc:
A004736 := proc(n,k)
        if n <0 or k < 1 or k > n then
                0;
        else
                n-k+1 ;
        end if;
end proc:
A158841 := proc(n,k)
        add( A145677(n-1,j-1)*A004736(j,k),j=k..n) ;
end proc: # R. J. Mathar, Nov 05 2011

Formula

T(n,k) = Sum_{j=k..n} A145677(n-1,j-1)*A004736(j,k), assuming column enumeration k >= 1 in A004736. - R. J. Mathar, Nov 05 2011

A158946 Triangle read by rows, A000012(signed) * A145677 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 3, 1, 4, 3, 3, 2, 4, 1, 5, 4, 3, 4, 2, 5, 1, 6, 4, 4, 3, 5, 2, 6, 1, 7, 5, 4, 5, 3, 6, 2, 7, 1, 8, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 6, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11
Offset: 1

Views

Author

Gary W. Adamson, Mar 31 2009

Keywords

Comments

Row sums = A000982 starting with offset 1: (1, 2, 5, 8, 13, 18, 25,...).

Examples

			First few rows of the triangle =
1;
1, 1;
2, 1, 2;
2, 2, 1, 3;
3, 2, 3, 1, 4;
3, 3, 2, 4, 1, 5;
4, 3, 4, 2, 5, 1, 6;
4, 4, 3, 5, 2, 6, 1, 7;
5, 4, 5, 3, 6, 2, 7, 1, 8;
5, 5, 4, 6, 3, 7, 2, 8, 1, 9;
6, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10;
6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11;
7, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12;
7, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13;
...

Crossrefs

Cf. A145677, A000982

Formula

Triangle read by rows, A000012(signed) * A145677 * A000012. A000012(signed) = an infinite lower triangular matrix with (1,-1,1,-1,...) in every column. A145677 = an infinite lower triangular matrix with all 1's as the left border, right border = (1, 1, 2, 3, 4, 5,...), and the rest zeros.

A158821 Triangle read by rows: row n (n>=0) ends with 1, and for n>=1 begins with n; other entries are zero.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 7, 0, 0, 0, 0, 0, 0, 1, 8, 0, 0, 0, 0, 0, 0, 0, 1, 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gary W. Adamson, Mar 30 2008

Keywords

Examples

			Triangle begins:
  1;
  1, 1;
  2, 0, 1;
  3, 0, 0, 1;
  4, 0, 0, 0, 1;
  5, 0, 0, 0, 0, 1;
  6, 0, 0, 0, 0, 0, 1;
  7, 0, 0, 0, 0, 0, 0, 1;

Links

Crossrefs

Cf. A000027, A109613, A145677.

Programs

  • Maple
    A158821:= proc(n,k)
    if n = k then 1;
    elif k = 0 then n;
    else 0;
    end if;
end proc: # R. J. Mathar, Jan 08 2015
  • Mathematica
    T[n_, k_]:= If[k==0, Boole[n==0] +n, If[k==n, 1, 0]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)
Join[{1},Table[Join[{n},PadLeft[{1},n,0]],{n,15}]]//Flatten (* Harvey P. Dale, Apr 05 2023 *)
  • Sage
    def A158821(n,k):
    if (k==0): return n + bool(n==0)
    elif (k==n): return 1
    else: return 0
flatten([[A158821(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

Formula

T(n, k) = A145677(n, n-k-1). - R. J. Mathar, Apr 01 2009
From G. C. Greubel, Dec 22 2021: (Start)
Sum_{k=0..n} T(n, k) = A000027(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A109613(n). (End)
Showing 1-4 of 4 results.

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