A327623 -id:A327623 - cp's OEIS frontend

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.

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, 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, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327632.

			Square array A(n,k) begins:
  0,  0,  0,   0,    0,    0,    0,     0,     0, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  3,  5,   7,    9,   11,   13,    15,    17, ...
  1,  3,  8,  16,   27,   41,   58,    78,   101, ...
  1,  5, 15,  35,   69,  121,  195,   295,   425, ...
  1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
  1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
  1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A057427, A015723, A327605, A327628.
Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.
Main diagonal gives: A327623.
Cf. A327618, A327632.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
    end:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

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]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).

A327619 Number of parts in all n-times partitions of n.

Original entry on oeis.org

0, 1, 5, 25, 219, 1596, 19844, 208377, 3394835, 46799236, 886886076, 15668835975, 366602236558, 7582277939549, 199035634246870, 4962275379320665, 150339081311823341, 4214812414260868163, 141823733752997729872, 4533014863242019822308, 169587948261109794026999

Offset: 0

Alois P. Heinz, Sep 19 2019

nonn

			a(2) = 5: 2, 11, 1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition (number theory)

Main diagonal of A327618.

Cf. A306187, A327623.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
         (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=0..21);

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]]]];
a[n_] := b[n, n, n][[2]];
a /@ Range[0, 21] (* Jean-François Alcover, May 01 2020, after Maple *)

cp's OEIS Frontend

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.

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