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A213191
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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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20
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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
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OFFSET
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0,6
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COMMENTS
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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
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LINKS
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FORMULA
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A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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