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 1-5 of 5 results.

A326616 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 5, 1, 9, 13, 9, 44, 42, 10, 96, 225, 150, 9, 152, 680, 1098, 576, 3, 155, 1350, 4155, 5201, 2266, 124, 2180, 11730, 26642, 26904, 9966, 140, 3751, 30300, 106281, 182000, 149832, 47466, 160, 6050, 69042, 348061, 896392, 1229760, 855240, 237019
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2019

Keywords

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

Examples

			T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...

Links

Crossrefs

Main diagonal gives A178682.
Row sums give A326648.
Column sums give A326650.
Cf. A000203, A185283, A326617 (this triangle read by columns), A326649, A326651.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++,  If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,   b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]][i*j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Feb 27 2021, after Alois P. Heinz *)

Formula

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).
Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

A244580 Square array read by antidiagonals related to the symmetric representation of sigma.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jul 04 2014

Keywords

Comments

The number of parts k in the square array is equal to A000203(k) hence the sum of parts k is equal to A064987(k).
The structure has a three-dimensional representation using polycubes. T(n,k) is the height of a column. The total area in the horizontal level z gives A000203(z).
The main diagonal gives A244367.

Examples

			.                         _ _ _ _ _ _ _ _ _
1,2,3,4,5,6,7,8,9...     |_| | | | | | | | |
2,2,3,4,5,6,7,8,9...     |_ _|_| | | | | | |
3,3,4,4,5,6,7,8,9...     |_ _|  _|_| | | | |
4,4,4,6,6,6,7,8,9...     |_ _ _|    _|_| | |
5,5,5,6,6,8,8,8,9...     |_ _ _|  _|  _ _|_|
6,6,6,6,8,8,9...         |_ _ _ _|  _| |
7,7,7,7,8,9,9...         |_ _ _ _| |_ _|
8,8,8,8,8...             |_ _ _ _ _|
9,9,9,9,9...             |_ _ _ _ _|
.

Crossrefs

Cf. A000203, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A244367.

A326617 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

Original entry on oeis.org

1, 1, 2, 2, 1, 5, 9, 9, 10, 9, 3, 13, 44, 96, 152, 155, 124, 140, 160, 113, 48, 16, 4, 42, 225, 680, 1350, 2180, 3751, 6050, 7420, 6870, 5555, 5330, 6300, 6475, 5025, 3000, 1250, 250, 150, 1098, 4155, 11730, 30300, 69042, 127364, 188568, 249690, 365160, 584733
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2019

Keywords

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

Examples

			T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...

Links

Crossrefs

Main diagonal gives A178682.
Row sums give A326648.
Column sums give A326650.
Cf. A000203, A024916, A326616 (this triangle read by rows), A326649, A326651.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), n=k..g(k)), k=0..6);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Formula

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).
Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

A326649 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

Original entry on oeis.org

0, 1, 4, 19, 81, 413, 2439, 14655, 86844, 573196, 4224230, 32280154, 249433713, 1925416359, 15732592327, 139542345546, 1304524118159, 12445507282579, 119198874300879, 1137647406084952, 11183828252431175, 116368970786569604, 1278400213028604214
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2019

Keywords

Links

Crossrefs

Cf. A185283, A326616, A326617.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..25);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return [k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || kJean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=A185283(n)..n} k * A326616(n,k) = Sum_{k=A185283(n)..n} k * A326617(n,k).

A245100 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n), multiplied by n.

Original entry on oeis.org

1, 6, 6, 6, 28, 15, 15, 72, 28, 28, 120, 45, 27, 45, 90, 90, 66, 66, 336, 91, 91, 168, 168, 120, 120, 120, 496, 153, 153, 702, 190, 190, 840, 231, 105, 105, 231, 396, 396, 276, 276, 1440, 325, 125, 325, 546, 546, 378, 162, 162, 378, 1568, 435, 435, 2160, 496, 496, 2016
Offset: 1

Views

Author

Omar E. Pol, Jul 11 2014

Keywords

Comments

Row sums give A064987.
Since both A000203(n) and A024916(n) have a symmetric representation then both row n and the triangle have can be represented as a symmetric polycube.

Examples

			The irregular triangle begins:
1;
6;
6, 6;
28;
15, 15;
72;
28, 28;
120;
45, 27, 45;
90, 90;
66, 66;
336;
91, 91;
168, 168;
120, 120, 120;
496;
153, 153;
702;
190, 190;
840;
231, 105, 105, 231;
...
For n = 9 the parts of the symmetric representation of sigma(9) are [5, 3, 5], so row 9 is [45, 27, 45].

Crossrefs

Cf. A000203, A024916, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A244580.

Formula

T(n,k) = n*A237270(n,k).
Showing 1-5 of 5 results.

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