A326650 -id:A326650 - cp's OEIS frontend

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.

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

Alois P. Heinz, Sep 12 2019

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

			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;
               ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

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

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);

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 *)

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

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

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

Alois P. Heinz, Sep 12 2019

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

			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;
               ...

Alois P. Heinz, Columns k = 0..40, flattened
Wikipedia, Partition (number theory)

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

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);

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 *)

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

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula