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.

A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Feb 14 2012

Keywords

Comments

Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012

Examples

			Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the six sections of 6):
.                                         6
.                                         3 3
.                                         4 2
.                                         2 2 2
.                            5              1
.                            3 2              1
.                  4           1              1
.                  2 2           1              1
.          3         1           1              1
.     2      1         1           1              1
.  1    1      1         1           1              1
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2,    1;
3,    1,  1;
6,    3,  1,  1;
8,    3,  2,  1,  1;
15,   8,  4,  2,  1,  1;
19,   8,  5,  3,  2,  1,  1;
32,  17,  9,  6,  3,  2,  1,  1;
42,  20, 13,  7,  5,  3,  2,  1,  1;
64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;

Crossrefs

Columns (1-2): A138137, A138135.
Row sums give A138879.
Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

Formula

From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)

Extensions

More terms from Alois P. Heinz, Feb 17 2012

A210947 Triangle read by rows: T(n,k) = total number of parts <= k of all partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 12, 16, 18, 19, 20, 19, 27, 31, 33, 34, 35, 30, 41, 47, 50, 52, 53, 54, 45, 64, 73, 79, 82, 84, 85, 86, 67, 93, 108, 116, 121, 124, 126, 127, 128, 97, 138, 159, 172, 180, 185, 188, 190, 191, 192
Offset: 1

Views

Author

Omar E. Pol, May 01 2012

Keywords

Comments

Row n lists the partial sums of row n of triangle A066633.

Examples

			Triangle begins:
1;
2,   3;
4,   5,  6;
7,  10,  11,  12;
12, 16,  18,  19,  20;
19, 27,  31,  33,  34,  35;
30, 41,  47,  50,  52,  53,  54;
45, 64,  73,  79,  82,  84,  85,  86;
67, 93, 108, 116, 121, 124, 126, 127, 128;

Links

Crossrefs

Column 1 is A000070(n-1). Right border gives A006128.
Cf. A066633, A181187, A206563, A210948, A210955.

Programs

  • Maple
    p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1]
    elif i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p (p (f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n, k) option remember;
       b(n, n)[k+1] +`if`(k<2, 0, T(n, k-1))
    end:
seq (seq (T(n,k), k=1..n), n=1..11); # Alois P. Heinz, May 02 2012
  • Mathematica
    p[f_, g_] := With[{m = Max[Length[f], Length[g]]}, PadRight[f, m, 0] + PadRight[g, m, 0]]; b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0, {1}, If[i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0}, b[n-i, i]]; p[p[f, g], Append[Array[0&, i], g[[1]] ]]]]]; T[n_, k_] := T[n, k] = b[n, n][[k+1]] + If[k<2, 0, T[n, k-1]]; Table [Table [T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{j=1..k} A066633(n,j).

A210956 Triangle read by rows: T(n,k) = sum of all parts <= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 3, 2, 2, 5, 3, 7, 7, 11, 5, 7, 10, 10, 15, 7, 15, 21, 25, 25, 31, 11, 17, 23, 27, 32, 32, 39, 15, 31, 40, 52, 57, 63, 63, 71, 22, 36, 54, 62, 72, 78, 85, 85, 94, 30, 60, 78, 98, 113, 125, 132, 140, 140, 150, 42, 72, 102, 122, 142, 154, 168, 176, 185, 185, 196
Offset: 1

Views

Author

Omar E. Pol, May 01 2012

Keywords

Comments

Row n lists the partial sums of row n of triangle A207383.

Examples

			Triangle begins:
1;
1,   3;
2,   2, 5;
3,   7, 7, 11;
5,   7, 10, 10, 15;
7,  15, 21, 25, 25, 31;
11, 17, 23, 27, 32, 32, 39;
15, 31, 40, 52, 57, 63, 63, 71;
22, 36, 54, 62, 72, 78, 85, 85, 94;

Links

Crossrefs

Column 1 is A000041. Right border gives A138879.
Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, 207383, 208476, A210948, A210955.

Programs

  • PARI
    Row(n)={my(v=vector(n)); v[1]=numbpart(n-1); if(n>1, forpart(p=n, for(k=1, #p, v[p[k]]++), [2,n])); for(k=2, n, v[k]=v[k-1]+k*v[k]); v}
{ for(n=1, 10, print(Row(n))) }

Formula

T(n,k) = Sum_{j=1..k} A207383(n,j).

Extensions

Terms a(46) and beyond from Andrew Howroyd, Feb 19 2020
Showing 1-3 of 3 results.

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