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A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A327622 #24 Dec 09 2020 11:05:12
%S A327622 0,0,1,0,1,1,0,1,1,1,0,1,1,3,1,0,1,1,5,3,1,0,1,1,7,8,5,1,0,1,1,9,16,
%T A327622 15,8,1,0,1,1,11,27,35,28,10,1,0,1,1,13,41,69,73,49,13,1,0,1,1,15,58,
%U A327622 121,160,170,86,18,1,0,1,1,17,78,195,311,460,357,156,25,1
%N A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A327622 Row n is binomial transform of the n-th row of triangle A327632.
%H A327622 Alois P. Heinz, <a href="/A327622/b327622.txt">Antidiagonals n = 0..200, flattened</a>
%H A327622 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327622 A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).
%e A327622 Square array A(n,k) begins:
%e A327622   0,  0,  0,   0,    0,    0,    0,     0,     0, ...
%e A327622   1,  1,  1,   1,    1,    1,    1,     1,     1, ...
%e A327622   1,  1,  1,   1,    1,    1,    1,     1,     1, ...
%e A327622   1,  3,  5,   7,    9,   11,   13,    15,    17, ...
%e A327622   1,  3,  8,  16,   27,   41,   58,    78,   101, ...
%e A327622   1,  5, 15,  35,   69,  121,  195,   295,   425, ...
%e A327622   1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
%e A327622   1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
%e A327622   1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...
%p A327622 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327622      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p A327622          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327622         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p A327622     end:
%p A327622 A:= (n, k)-> b(n$2, k)[2]:
%p A327622 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A327622 b[n_, i_, k_] := b[n, i, k] = With[{}, If[n==0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i(i + 1)/2 < n, Return@{0, 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 - 1], k]]][b[i, i, k - 1]]];
%t A327622 A[n_, k_] := b[n, n, k][[2]];
%t A327622 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jun 03 2020, after Maple *)
%Y A327622 Columns k=0-3 give: A057427, A015723, A327605, A327628.
%Y A327622 Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.
%Y A327622 Main diagonal gives: A327623.
%Y A327622 Cf. A327618, A327632.
%K A327622 nonn,tabl
%O A327622 0,14
%A A327622 _Alois P. Heinz_, Sep 19 2019