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.

A125105 Triangular array with the first half of the odd-indexed rows of A048004.

Original entry on oeis.org

1, 1, 4, 1, 12, 11, 1, 33, 47, 27, 1, 88, 185, 127, 63, 1, 232, 694, 563, 303, 143, 1, 609, 2526, 2400, 1394, 687, 319, 1, 1596, 9012, 9960, 6215, 3186, 1519, 703, 1, 4180, 31709, 40534, 27095, 14401, 7026, 3311, 1535, 1, 10945, 110469, 162538, 116143, 63872, 31808, 15218, 7151, 3327
Offset: 1

Views

Author

Alford Arnold, Dec 07 2006

Keywords

Comments

A000079 counts compositions admitting a variety of triangular views; for example, A048004 and A105147. The subtable formed from the odd rows of A048004 has row sums 1, 8, 44, 208, 912, ... . Because only the first half of rows of A048004 is transferred to this triangle here, there is a difference between row sums of A048004 and row sums here, A045623(n-1).

Examples

			The odd-indexed rows of triangle A048004 begin
  1  1
  1  4  2 1
  1 12 11 5 2 1
  ...
so the triangle here begins
  1
  1  4
  1 12 11
  ...

Crossrefs

Cf. A008464 (row sums), A008464, A045623, A048004, A105147, A125104.

Programs

  • Maple
    A048004 := proc(n,k) option remember ; if k < 0 or k > n then 0; elif k = 0 or k = n then 1; else 2*procname(n-1,k)+procname(n-1,k-1)-2*procname(n-2,k-1)+procname(n-k-1,k-1)-procname(n-k-2,k) ; fi ; end:
A125105 := proc(n,k) A048004(2*n-1,k) ; end:
for n from 1 to 13 do for k from 0 to n-1 do printf("%d ",A125105(n,k)) ; od: od: # R. J. Mathar, Nov 23 2007
  • Mathematica
    B[n_, k_] := B[n, k] = If[n == 0 || k == 1, 1, Sum[B[n - j, k], {j, 1, Min[n, k]}]];
A048004[n_, k_] := B[n + 1, k + 1] - B[n + 1, k];
T[n_, k_] := A048004[2 n - 1, k];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 27 2024, after Maple code here and in A048004 *)

Formula

T(n,k) = A048004(2*n-1,k), 0 <= k < n. - R. J. Mathar, Nov 23 2007

Extensions

More terms from R. J. Mathar, Nov 23 2007

A200047 Number of compositions of n having smallest part equal to 2.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 5, 9, 15, 25, 42, 70, 116, 192, 317, 522, 858, 1408, 2307, 3775, 6170, 10074, 16433, 26784, 43623, 71002, 115497, 187777, 305147, 495669, 804839, 1306404, 2119884, 3438952, 5577396, 9043549, 14660810, 23762784, 38509220, 62397495, 101090631
Offset: 1

Views

Author

Alois P. Heinz, Nov 13 2011

Keywords

Examples

			a(7) = 5: [2,2,3], [2,3,2], [2,5], [3,2,2], [5,2].

Links

Crossrefs

Column k=2 of A105147.

Programs

  • Maple
    a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>, <-1|-1|0|0|2>>^n. <<0, 0, 1, 0, 1>>)[1, 1]: seq (a(n), n=1..50);

Formula

G.f.: (x-1)^2*x^2/(1-2*x+x^4+x^5).

A125104 Triangle read by rows counting compositions (ordered partitions) by minimal part size.

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 1, 6, 1, 0, 0, 2, 13, 1, 0, 0, 1, 3, 27, 1, 0, 0, 0, 2, 5, 56, 1, 0, 0, 0, 1, 2, 9, 115, 1, 0, 0, 0, 0, 2, 3, 15, 235, 1, 0, 0, 0, 0, 1, 2, 5, 25, 478, 1, 0, 0, 0, 0, 0, 2, 2, 8, 42, 969, 1, 0, 0, 0, 0, 0, 1, 2, 3, 12, 70, 1959, 1, 0, 0, 0, 0, 0, 0, 2, 2, 5, 18, 116, 3952, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 8, 27, 192, 7959, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 11, 41, 317, 16007
Offset: 0

Views

Author

Alford Arnold, Nov 28 2006, corrected Nov 28 2006

Keywords

Comments

The diagonals of this array can be generated from Table A099238 as follows: A000079 - A000045 = [1, 2, 4, 8, 16, 32, ...] - [0, 1, 1, 2, 3, 5, ...] = [1, 1, 3, 6, 13, 27, ...] = A099036, A000045 - A000930, A000930 - A003269, A003269 - A003520, etc.

Examples

			Row 4 of the array is (1, 0, 1, 6) because there are six compositions with minimum part of size one: 1111, 31, 13, 211, 121, 112; one of size two: 22; none of size three; and 1 of size four: 4.
Triangle (after 45-degree counterclockwise rotation) begins:
1 1 3 6 13 27 56 115 235 478 969 1959 3952 7959
.1 0 1 2 3 5 9 15 25 42 70 116 192
..1 0 0 1 2 2 3 5 8 12 18 27
...1 0 0 0 1 2 2 2 3 5 8
....1 0 0 0 0 1 2 2 2 2
.....1 0 0 0 0 0 1 2 2
......1 0 0 0 0 0 0 1
.......1 0 0 0 0 0 0
........1 0 0 0 0 0

Crossrefs

Cf. A000079, A000045, A000930, A003269, A003520, A099036, A099238.
Cf. A105147.

Extensions

Edited by N. J. A. Sloane, Dec 21 2006
More terms from Vladeta Jovovic, Jul 10 2007
Showing 1-3 of 3 results.

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

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