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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

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