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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 (list; table; graph; refs; listen; history; text; internal format)
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

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

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

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.

Sequence in context: A220421 A106683 A139601 * A079520 A229001 A208981

Adjacent sequences:  A213188 A213189 A213190 * A213192 A213193 A213194

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 28 2013

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)