A320739 -id:A320739 - 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.

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).

A320822 Number of partitions of n with exactly nine sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 45, 1156, 22321, 360704, 5158652, 67512671, 826325926, 9601854284, 107074391802, 1155110296163, 12128784832510, 124544257464575, 1255354199998891, 12457896065009874, 122014591067156240, 1181759128040203056, 11337243349606910986, 107879491494316487583

Offset: 9

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1053

Column k=9 of A292746.

Cf. A320739, A320740.

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))(9):
seq(a(n), n=9..35);

a(n) = A320740(n) - A320739(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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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

A320822 Number of partitions of n with exactly nine sorts of part 1 which are introduced in ascending order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula