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%I #26 Apr 30 2020 07:24:44
%S 0,0,1,0,1,1,0,1,3,1,0,1,5,6,1,0,1,7,14,12,1,0,1,9,25,44,20,1,0,1,11,
%T 39,109,100,35,1,0,1,13,56,219,315,274,54,1,0,1,15,76,386,769,1179,
%U 581,86,1,0,1,17,99,622,1596,3643,3234,1417,128,1,0,1,19,125,939,2960,9135,12336,10789,2978,192,1
%N Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Row n is binomial transform of the n-th row of triangle A327631.
%H Alois P. Heinz, <a href="/A327618/b327618.txt">Antidiagonals n = 0..200, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A(n,k) = Sum_{i=0..k} binomial(k,i) * A327631(n,i).
%e A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 3, 5, 7, 9, 11, 13, 15, ...
%e 1, 6, 14, 25, 39, 56, 76, 99, ...
%e 1, 12, 44, 109, 219, 386, 622, 939, ...
%e 1, 20, 100, 315, 769, 1596, 2960, 5055, ...
%e 1, 35, 274, 1179, 3643, 9135, 19844, 38823, ...
%e 1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...
%p b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p end:
%p A:= (n, k)-> b(n$2, k)[2]:
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
%t A[n_, k_] := b[n, n, k][[2]];
%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-3 give: A057427, A006128, A327594, A327627.
%Y Rows n=0-3 give: A000004, A000012, A005408, A095794(k+1).
%Y Main diagonal gives A327619.
%Y Cf. A323718, A327622, A327631.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Sep 19 2019
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