A327631 -id:A327631 - cp's OEIS frontend

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

1, 1, 1, 3, 6, 24, 84, 498, 2220, 15108, 92328, 773580, 5636460, 53563476, 471562512, 5270698716, 52117937052, 637276396764, 7317811499736, 100453675122444, 1276319138168796, 19048874583061716, 270233458572751440, 4442429353548965628, 68384217440167826412

Offset: 1

Alois P. Heinz, Sep 20 2019

nonn

Number of proper (n-1)-times partitions of n, cf. A327639.
Might be called "Half-Factorial numbers" analog to the "Half-Catalan numbers" (A000992).
The recursion formula is a special case of the formula given in A327729.
a(n+1)/(n*a(n)) tends to 0.67617164... - Vaclav Kotesovec, Apr 28 2020

			a(1) = 1:
  1
a(2) = 1:
  2 -> 11
a(3) = 1:
  3 -> 21 -> 111
a(4) = 3:
  4 -> 31 -> 211 -> 1111
  4 -> 22 -> 112 -> 1111
  4 -> 22 -> 211 -> 1111
a(5) = 6:
  5 -> 41 -> 311 -> 2111 -> 11111
  5 -> 41 -> 221 -> 1121 -> 11111
  5 -> 41 -> 221 -> 2111 -> 11111
  5 -> 32 -> 212 -> 1112 -> 11111
  5 -> 32 -> 212 -> 2111 -> 11111
  5 -> 32 -> 311 -> 2111 -> 11111

Alois P. Heinz, Table of n, a(n) for n = 1..481
Vaclav Kotesovec, Plot of a(n+1)/(n*a(n)) for n = 1..10000
Wikipedia, Partition (number theory)

Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
seq(a(n), n=1..29);
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
    end:
seq(a(n), n=1..29);

a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
Array[a, 25] (* Jean-François Alcover, Apr 28 2020 *)

a(n) = Sum_{j=1..floor(n/2)} C(n-2,j-1) a(j)*a(n-j) for n > 1, a(1) = 1.

a(n) = A327639(n,n-1) = A327631(n,n-1)/n.

A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 15, 16, 6, 1, 10, 45, 88, 76, 24, 1, 14, 93, 282, 420, 302, 84, 1, 21, 223, 1052, 2489, 3112, 1970, 498, 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220, 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108

Offset: 0

Alois P. Heinz, Sep 20 2019

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A323718.

			T(4,0) = 1:  4
T(4,1) = 4:     T(4,2) = 6:          T(4,3) = 3:
  4-> 31          4-> 31 -> 211        4-> 31 -> 211 -> 1111
  4-> 22          4-> 31 -> 1111       4-> 22 -> 112 -> 1111
  4-> 211         4-> 22 -> 112        4-> 22 -> 211 -> 1111
  4-> 1111        4-> 22 -> 211
                  4-> 22 -> 1111
                  4-> 211-> 1111
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2,   1;
  1,  4,   6,    3;
  1,  6,  15,   16,     6;
  1, 10,  45,   88,    76,     24;
  1, 14,  93,  282,   420,    302,     84;
  1, 21, 223, 1052,  2489,   3112,   1970,    498;
  1, 29, 444, 2950,  9865,  18123,  18618,  10046,  2220;
  1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
  ...

Alois P. Heinz, Rows n = 0..170, flattened
Wikipedia, Iverson bracket
Wikipedia, Partition (number theory)

Columns k=0-2 give A000012, A000065, A327769.
Row sums give A327644.
T(2n,n) gives A327645.
Cf. A323718, A327631, A327643, A327646.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[0, n - 1] }] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A323718(n,i).
T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.
Sum_{k=1..n-1} k * T(n,k) = A327646(n).
Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.

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.

Original entry on oeis.org

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).

A327632 Number T(n,k) of parts in all proper k-times partitions of n into distinct parts; triangle T(n,k), n >= 1, 0 <= k <= max(0,n-2), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 4, 6, 4, 1, 7, 13, 12, 5, 1, 9, 30, 52, 35, 6, 1, 12, 61, 137, 156, 72, 7, 1, 17, 121, 384, 638, 548, 196, 8, 1, 24, 210, 880, 1983, 2442, 1543, 400, 9, 1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10, 1, 39, 600, 4477, 17883, 40855, 54279, 40511, 15178, 2070, 11

Offset: 1

Alois P. Heinz, Sep 19 2019

In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A327622.

			T(4,0) = 1:
  4    (1 part).
T(4,1) = 2:
  4-> 31    (2 parts)
T(4,2) = 3:
  4-> 31 -> 211   (3 parts)
Triangle T(n,k) begins:
  1;
  1;
  1,  2;
  1,  2,   3;
  1,  4,   6,    4;
  1,  7,  13,   12,    5;
  1,  9,  30,   52,   35,     6;
  1, 12,  61,  137,  156,    72,     7;
  1, 17, 121,  384,  638,   548,   196,    8;
  1, 24, 210,  880, 1983,  2442,  1543,  400,    9;
  1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10;
  ...

Alois P. Heinz, Rows n = 1..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-2 give: A057427, -1+A015723(n), A327795.
Row sums give A327647.
Cf. A327622, A327631.

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:
T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-2)), n=1..14);

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {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[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, Max[0, n - 2]}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327622(n,i).

T(n+1,n-1) = 1 for n >= 1.

A327648 Number of parts in all proper many times partitions of n.

Original entry on oeis.org

0, 1, 3, 9, 45, 185, 1277, 7469, 67993, 514841, 5414197, 52609653, 679432169, 7704502013, 111283754969, 1515535050805, 25257251330321, 385282195339393, 7088110874426409, 123325149268482781, 2520808658222616653, 48623257343586890769, 1078165538033926164281

Offset: 0

Alois P. Heinz, Sep 20 2019

nonn

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

			a(3) = 9 = 1 + 2 + 3 + 3, counting the (final) parts in: 3, 3->21, 3->111, 3->21->111.
a(4) = 45: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111.

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

Row sums of A327631.

Cf. A327644, A327647.

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-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..n-1):
seq(a(n), n=0..25);

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_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {k, 0, n - 1}, {i, 0, k}];
a /@ Range[0, 25] (* Jean-François Alcover, May 01 2020, after Maple *)

A328042 Number of parts in all proper twice partitions of n.

Original entry on oeis.org

3, 21, 61, 205, 474, 1246, 2723, 6277, 12961, 28682, 56976, 118919, 234715, 473988, 913011, 1807211, 3430048, 6648397, 12500170, 23764885, 44174088, 83090853, 152803509, 283387971, 517516615, 949775754, 1719088271, 3127641937, 5618833687, 10133255636

Offset: 3

Alois P. Heinz, Oct 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 3..5000 (terms 3..2000 from Alois P. Heinz)

Column k=2 of A327631.

Cf. A327769.

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-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=3..35);

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_] := With[{k = 2}, Sum[b[n, n, i][[2]] (-1)^(k-i) Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 35] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A328041 Number of parts in all proper floor(n/2)-times partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 21, 61, 461, 1652, 17487, 76264, 1002835, 5207742, 88664398, 515821495, 10184805624, 69200406679, 1610282904928, 12024183111167, 318978837371853, 2653055962437988, 79332250069994262, 725413309833320933, 23919660963588169669, 238830233430136549070

Offset: 0

Alois P. Heinz, Oct 02 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. 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-> add(b(n$2, i)[2]*(-1)^(k-i)
        *binomial(k, i), i=0..k))(iquo(n,2)):
seq(a(n), n=0..23);

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_] := With[{k = Quotient[n, 2]}, Sum[b[n, n, i][[2]] (-1)^(k - i)* Binomial[k, i], {i, 0, k}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(n) = A327631(n,floor(n/2)).

cp's OEIS Frontend

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

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A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

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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.

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A327632 Number T(n,k) of parts in all proper k-times partitions of n into distinct parts; triangle T(n,k), n >= 1, 0 <= k <= max(0,n-2), read by rows.

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A327648 Number of parts in all proper many times partitions of n.

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A328042 Number of parts in all proper twice partitions of n.

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A328041 Number of parts in all proper floor(n/2)-times partitions of n.

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