A134869 -id:A134869 - cp's OEIS frontend

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 5, 1, 1, 0, 1, 7, 4, 2, 1, 0, 1, 11, 3, 2, 1, 1, 0, 1, 15, 8, 3, 3, 1, 1, 0, 1, 22, 7, 6, 2, 2, 1, 1, 0, 1, 30, 15, 6, 5, 3, 2, 1, 1, 0, 1, 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1, 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Nov 28 2010

For the definition of "section" of the set of partitions of n see A135010.
Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.
From Omar E. Pol, Feb 12 2012: (Start)
It appears that reversed rows converge to A002865.
It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:
.
.         1,
.      1, 0, 1,
.   4, 2, 1, 0, 1,
11, 3, 2, 1, 1, 0, 1,
---------------------
11, 7, 5, 3, 2, 1, 1,
.
It appears that in row n starts an infinite trapezoid in which column sums always give the number of partitions of n-1. Example for n = 7:
.
11, 3, 2, 1, 1, 0, 1,
.   8, 3, 3, 1, 1, 0, 1,
.      6, 2, 2, 1, 1, 0, 1,
.         5, 3, 2, 1, 1, 0, 1,
.            4, 2, 2, 1, 1, 0, 1,
.               5, 2, 2, 1, 1, 0,...
.                  4, 2, 2, 1, 1,...
.                     4, 2, 2, 1,...
.                        4, 2, 2,...
.                           4, 2,...
.                              4,...
.
The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:
11,
.    8,
.    7,  6,
.        6,  5,
.       10,  5, ...
.           10, ...
.           10, ...
-------------------
11, 15, 22, 30, ...
(End)
It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012
More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013

			Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of this triangle.
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1.
Triangle begins:
   1;
   1,  1;
   2,  0,  1;
   3,  2,  0,  1;
   5,  1,  1,  0, 1;
   7,  4,  2,  1, 0, 1;
  11,  3,  2,  1, 1, 0, 1;
  15,  8,  3,  3, 1, 1, 0, 1;
  22,  7,  6,  2, 2, 1, 1, 0, 1;
  30, 15,  6,  5, 3, 2, 1, 1, 0, 1;
  42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1;
  56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Omar E. Pol, Illustration of the section model of partitions, Figure 1 (n = 1..6), Figure 2 (2D view, n = 1..10), Figure 3 (3D view, n = 1..9)

Row sums give A138137. Where records occur is A134869.
Columns 1-10: A000041, A182712-A182714, A206555-A206560.
Sub-triangles (1-11): A023531, A129186, A194702-A194710
Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n,i) option remember; local g;
      if n=0        then [1]
    elif n<2 or i<2 then [0]
    else g:=   `if`(i>n, [0],  b(n-i, i));
         p(p([0$j=2..i, g[1]], b(n, i-1)), g)
      fi
    end:
h:= proc(n) option remember;
      `if`(n=0, 1, b(n, n)[1]+h(n-1))
    end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 19 2012

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k,2,n}], {n,1,12}]  // Flatten (* Robert Price, May 15 2020 *)

It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012

A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1. For n >= 2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is odd and k=n/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 4, 1, 1, 6, 16, 11, 5, 1, 1, 7, 22, 27, 16, 6, 1, 1, 8, 29, 65, 43, 22, 7, 1, 1, 9, 37, 94, 108, 65, 29, 8, 1, 1, 10, 46, 131, 267, 173, 94, 37, 9, 1, 1, 11, 56, 177, 398, 440, 267, 131, 46, 10, 1, 1, 12, 67, 233

Offset: 0

Clark Kimberling

T(n+2,n) = A134869(n+1). - Philippe Deléham, Feb 01 2014

			Triangle begins:
1
1  1
1  2  1
1  4  3   1
1  5  7   4   1
1  6 16  11   5    1
1  7 22  27  16    6   1
1  8 29  65  43   22   7   1
1  9 37  94 108   65  29   8   1
1 10 46 131 267  173  94  37   9  1
1 11 56 177 398  440 267 131  46 10  1
1 12 67 233 575 1105 707 398 177 56 11 1
... - _Philippe Deléham_, Feb 01 2014

G. C. Greubel, Rows n = 0..99 of triangle, flattened
Rob Arthan, Comments on A026674, A026725, A026670

Cf. A026674.

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
Flat(List([0..14], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 16 2019

A026725 := proc(n,k)
    option remember;
    if n < 0 or k < 0 then
        0;
    elif k=0 or k=n then
        1;
    elif 2*k = n-1 then
      procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 21 2019

T[n_, k_]:= T[n, k]= If[k==0||k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 16 2019 *)

T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
for(n=0,11, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 16 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jul 16 2019

T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, i+1)-to-(i+1, i+2) for i >= 0.

Comment from Rick L. Shepherd, Aug 05 2002: Probably this should be changed to "and edges (i+1, i)-to-(i+2, i+1) for i >= 0."

Title and offset corrected by G. C. Greubel, Jul 16 2019, again by R. J. Mathar, Oct 21 2019, again by Sean A. Irvine, Oct 25 2019

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Original entry on oeis.org

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Reinhard Zumkeller, Jul 17 2014

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A046092, A062725, A245301.

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)

A134868 A103451 * A002260.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 14 2007

Row sums = A134869: (1, 4, 7, 11, 16, 22, 29, ...).

			First few rows of the triangle:
  1;
  2, 2;
  2, 2, 3;
  2, 2, 3, 4;
  2, 2, 3, 4, 5;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002260, A071324, A134869.

Table[k + Boole[k == 1 && n != 2], {n, 2, 14}, {k, n - 1}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

A103451 * A002260 as infinite lower triangular matrices.

Left border of A002260, (1, 1, 1, 1, ...) is replaced by (1, 2, 2, 2, ...).

cp's OEIS Frontend

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1. For n >= 2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is odd and k=n/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Sage

Formula

Extensions

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Sage

Formula

A134868 A103451 * A002260.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula