A213180 -id:A213180 - cp's OEIS frontend

A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 6, 9, 12, 0, 1, 10, 17, 20, 20, 0, 1, 18, 39, 44, 35, 35, 0, 1, 34, 101, 122, 87, 66, 54, 0, 1, 66, 279, 392, 287, 180, 105, 86, 0, 1, 130, 797, 1370, 1119, 660, 311, 176, 128, 0, 1, 258, 2319, 5024, 4775, 2904, 1281, 558, 270, 192

Offset: 0

Alois P. Heinz, Feb 28 2013

In general, if k > 0 then column k is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - Vaclav Kotesovec, May 27 2018

			Square array A(n,k) begins:
:   0,  0,   0,   0,    0,     0,     0, ...
:   1,  1,   1,   1,    1,     1,     1, ...
:   3,  4,   6,  10,   18,    34,    66, ...
:   6,  9,  17,  39,  101,   279,   797, ...
:  12, 20,  44, 122,  392,  1370,  5024, ...
:  20, 35,  87, 287, 1119,  4775, 21447, ...
:  35, 66, 180, 660, 2904, 14196, 73920, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A006128, A066186, A066183, A229325, A229326, A229327, A229328, A229329, A229330, A229331, A229332.
Rows n=0-10 give: A000004, A000012, A052548, A229354, A229355, A229356, A229357, A229358, A229359, A229360, A229361.
Main diagonal gives A252761.
Cf. A213180.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
          (b(n-p*m, p-1, k)), m=0..n/p)))
    end:
A:= (n, k)-> b(n, n, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 0, l, l + {0, First[l]*p^k*m}]][b[n - p*m, p - 1, k]], { m, 0, n/p}]]] ; a[n_, k_] := b[n, n, k][[2]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; A[n_, k_] := Sum[T[n, j]*j^k, {j, 1, n}]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 15 2016 *)

A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.

A197126 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of cliques of size k in all partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 4, 2, 0, 1, 8, 2, 1, 0, 1, 11, 4, 2, 1, 0, 1, 19, 5, 3, 1, 1, 0, 1, 26, 10, 3, 3, 1, 1, 0, 1, 41, 11, 7, 3, 2, 1, 1, 0, 1, 56, 20, 8, 5, 3, 2, 1, 1, 0, 1, 83, 25, 13, 6, 5, 2, 2, 1, 1, 0, 1, 112, 38, 17, 11, 5, 5, 2, 2, 1, 1, 0, 1, 160, 49, 25, 13, 9, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 1

Alois P. Heinz, Oct 10 2011

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			T(4,1) = 4: [1,1,(2)], [(1),(3)], [(4)].
T(8,3) = 3: [1,1,(2,2,2)], [(1,1,1),2,3], [(1,1,1),5].
T(12,4) = 11: [(1,1,1,1),(2,2,2,2)], [1,(2,2,2,2),3], [(1,1,1,1),2,3,3], [(3,3,3,3)], [(1,1,1,1),2,2,4], [(2,2,2,2),4], [(1,1,1,1),4,4], [(1,1,1,1),3,5], [(1,1,1,1),2,6], [(1,1,1,1),8].  Here the first partition contains 2 cliques.
Triangle begins:
   1;
   1,  1;
   3,  0, 1;
   4,  2, 0, 1;
   8,  2, 1, 0, 1;
  11,  4, 2, 1, 0, 1;
  19,  5, 3, 1, 1, 0, 1;
  26, 10, 3, 3, 1, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Abdulaziz M. Alanazi and Augustine O. Munagi, On partition configurations of Andrews-Deutsch, Integers 17 (2017), #A7.

Columns k=1-10 give: A024786(n+1), A116646, A117524, A222704, A222705, A222706, A222707, A222708, A222709, A222710.
Row sums give: A000070(n-1). Diagonal gives: A000012.  Limit of reversed rows: T(2*n+1,n+1) = A002865(n).
Cf. A213180.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..20);

Table[CoefficientList[ 1/q* Tr[Flatten[q^Map[Length, Split /@ IntegerPartitions[n], {2}]]], q], {n, 24}] (* Wouter Meeussen, Apr 21 2012 *)
b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == k, l + {0, l[[1]]}, l]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

G.f. of column k: (x^k/(1-x^k)-x^(k+1)/(1-x^(k+1)))/Product_{j>0}(1-x^j).

Column k is asymptotic to exp(Pi*sqrt(2*n/3)) / (k*(k+1)*Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, May 24 2018

cp's OEIS Frontend

A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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