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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.
20
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
0,6
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
LINKS
FORMULA
A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.
EXAMPLE
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, ...
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<n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 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 *)
CROSSREFS
Main diagonal gives A252761.
Cf. A213180.
Sequence in context: A352493 A106683 A139601 * A352449 A375546 A079520
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 28 2013
STATUS
approved