A327619 -id:A327619 - cp's OEIS frontend

A327618 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.

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, 39, 109, 100, 35, 1, 0, 1, 13, 56, 219, 315, 274, 54, 1, 0, 1, 15, 76, 386, 769, 1179, 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

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327631.

			A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
Square array A(n,k) begins:
  0,  0,   0,    0,     0,     0,     0,      0, ...
  1,  1,   1,    1,     1,     1,     1,      1, ...
  1,  3,   5,    7,     9,    11,    13,     15, ...
  1,  6,  14,   25,    39,    56,    76,     99, ...
  1, 12,  44,  109,   219,   386,   622,    939, ...
  1, 20, 100,  315,   769,  1596,  2960,   5055, ...
  1, 35, 274, 1179,  3643,  9135, 19844,  38823, ...
  1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...

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

Columns k=0-3 give: A057427, A006128, A327594, A327627.
Rows n=0-3 give: A000004, A000012, A005408, A095794(k+1).
Main diagonal gives A327619.
Cf. A323718, A327622, A327631.

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, 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] = 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_, k_] := b[n, n, k][[2]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327631(n,i).

A306187 Number of n-times partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 65, 371, 3780, 33552, 472971, 5736082, 97047819, 1547576394, 32992294296, 626527881617, 15202246707840, 352290010708120, 9970739854456849, 262225912049078193, 8309425491887714632, 250946978120046026219, 8898019305511325083149

Offset: 0

Alois P. Heinz, Jan 27 2019

nonn

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - Gus Wiseman, Jan 27 2019

			From _Gus Wiseman_, Jan 27 2019: (Start)
The a(1) = 1 through a(3) = 10 partitions:
  (1)  ((2))     (((3)))
       ((11))    (((21)))
       ((1)(1))  (((111)))
                 (((2)(1)))
                 (((11)(1)))
                 (((2))((1)))
                 (((1)(1)(1)))
                 (((11))((1)))
                 (((1)(1))((1)))
                 (((1))((1))((1)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..410
Wikipedia, Partition (number theory)

Main diagonal of A323718.
Cf. A000041, A306188.
Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

ptnlevct[n_,k_]:=Switch[k,0,1,1,PartitionsP[n],_,SeriesCoefficient[Product[1/(1-ptnlevct[m,k-1]*x^m),{m,n}],{x,0,n}]];
Table[ptnlevct[n,n],{n,0,8}] (* Gus Wiseman, Jan 27 2019 *)

a(n) = A323718(n,n).

A327623 Number of parts in all n-times partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 1, 7, 27, 121, 553, 3865, 24625, 202954, 1519540, 14193455, 132441998, 1381539355, 14096067555, 168745220585, 1961128020387, 25473872598375, 324797436024684, 4647784901400988, 65394584337577858, 1012005650484163962, 15285115573675197704

Offset: 0

Alois P. Heinz, Sep 19 2019

nonn

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

Main diagonal of A327622.

Cf. A327619.

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-> b(n$3)[2]:
seq(a(n), n=0..23);

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_] := b[n, n, n][[2]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A327618 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A306187 Number of n-times partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A327623 Number of parts in all n-times partitions of n into distinct parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica