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

Author

Alois P. Heinz, Feb 28 2013

Keywords

• nonn
• tabl

Comments

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

Examples

			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, ...
		

Links

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

Crossrefs

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.

Programs

Maple

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);

Mathematica

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 *)

Formula

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