A320738 -id:A320738 - cp's OEIS frontend

A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 6, 5, 2, 1, 1, 3, 7, 13, 7, 4, 1, 1, 3, 7, 19, 26, 11, 4, 1, 1, 3, 7, 20, 52, 54, 15, 7, 1, 1, 3, 7, 20, 62, 151, 108, 22, 8, 1, 1, 3, 7, 20, 63, 217, 442, 219, 30, 12, 1, 1, 3, 7, 20, 63, 232, 803, 1314, 439, 42, 14

Offset: 0

Alois P. Heinz, Sep 22 2017

			A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,  1,   1,    1,    1,    1,    1,    1,    1, ...
  1,  2,   3,    3,    3,    3,    3,    3,    3, ...
  1,  3,   6,    7,    7,    7,    7,    7,    7, ...
  2,  5,  13,   19,   20,   20,   20,   20,   20, ...
  2,  7,  26,   52,   62,   63,   63,   63,   63, ...
  4, 11,  54,  151,  217,  232,  233,  233,  233, ...
  4, 15, 108,  442,  803,  944,  965,  966,  966, ...
  7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A002865, A000041, A320733, A320734, A320735, A320736, A320737, A320738, A320739, A320740, A320741.
Main diagonal gives A292503.
Cf. A292508, A292622, A292741, A292746.

f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)

A(n,k) = Sum_{j=0..k} A292746(n,j).

A(n,k) = A(n,n) for all k >= n.

A320820 Number of partitions of n with exactly seven sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 28, 463, 5909, 64479, 633796, 5786275, 50033463, 415225854, 3338335646, 26179143977, 201266007483, 1522856635641, 11374331041836, 84061202478127, 615860361908534, 4479596579257904, 32388729758708314, 233011769893620853, 1669336230635613631

Offset: 7

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1187

Column k=7 of A292746.

Cf. A320737, A320738.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(7):
seq(a(n), n=7..35);

a(n) = A320738(n) - A320737(n).

A320821 Number of partitions of n with exactly eight sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 36, 751, 11917, 159815, 1912316, 21084803, 218711887, 2164920950, 20657703246, 191440769945, 1732792167043, 15385193971985, 134455882817716, 1159708265019855, 9893526482067374, 83627808435796896, 701411197245083482, 5844301347854288709, 48423747013469923303

Offset: 8

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1112

Column k=8 of A292746.

Cf. A320738, A320739.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(8):
seq(a(n), n=8..35);

a(n) = A320739(n) - A320738(n).

cp's OEIS Frontend

A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A320820 Number of partitions of n with exactly seven sorts of part 1 which are introduced in ascending order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A320821 Number of partitions of n with exactly eight sorts of part 1 which are introduced in ascending order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula