A328042 -id:A328042 - cp's OEIS frontend

A327631 Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows.

1, 1, 2, 1, 5, 3, 1, 11, 21, 12, 1, 19, 61, 74, 30, 1, 34, 205, 461, 432, 144, 1, 53, 474, 1652, 2671, 2030, 588, 1, 85, 1246, 6795, 17487, 23133, 15262, 3984, 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980, 1, 191, 6277, 69812, 360114, 1002835, 1606434, 1483181, 734272, 151080

Offset: 1

Alois P. Heinz, Sep 19 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 A327618.

			T(4,0) = 1:
  4    (1 part).
T(4,1) = 11 = 2 + 2 + 3 + 4:
  4-> 31    (2 parts)
  4-> 22    (2 parts)
  4-> 211   (3 parts)
  4-> 1111  (4 parts)
T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4:
  4-> 31 -> 211   (3 parts)
  4-> 31 -> 1111  (4 parts)
  4-> 22 -> 112   (3 parts)
  4-> 22 -> 211   (3 parts)
  4-> 22 -> 1111  (4 parts)
  4-> 211-> 1111  (4 parts)
T(4,3) = 12 = 4 + 4 + 4:
  4-> 31 -> 211 -> 1111  (4 parts)
  4-> 22 -> 112 -> 1111  (4 parts)
  4-> 22 -> 211 -> 1111  (4 parts)
Triangle T(n,k) begins:
  1;
  1,   2;
  1,   5,    3;
  1,  11,   21,    12;
  1,  19,   61,    74,    30;
  1,  34,  205,   461,   432,    144;
  1,  53,  474,  1652,  2671,   2030,    588;
  1,  85, 1246,  6795, 17487,  23133,  15262,  3984;
  1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980;
  ...

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

Columns k=0-2 give: A057427, -1+A006128(n), A328042.
Row sums give A327648.
T(n,floor(n/2)) gives A328041.
Cf. A327618, A327632, A327639, A327643.

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:
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..n-1), n=1..12);

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]]]];
T[n_, k_] := Sum[b[n, n, i][[2]]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327618(n,i).

T(n,n-1) = n * A327639(n,n-1) = n * A327643(n) for n >= 1.

A327769 Number of proper twice partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 6, 15, 45, 93, 223, 444, 944, 1802, 3721, 6898, 13530, 25150, 48047, 87702, 165173, 298670, 553292, 995698, 1815981, 3242921, 5872289, 10406853, 18630716, 32879716, 58391915, 102371974, 180622850, 314943742, 551841083, 958011541, 1667894139

Offset: 0

Alois P. Heinz, Sep 24 2019

nonn

			a(3) = 1:
  3 -> 21 -> 111
a(4) = 6:
  4 -> 31 -> 211
  4 -> 31 -> 1111
  4 -> 22 -> 112
  4 -> 22 -> 211
  4 -> 22 -> 1111
  4 -> 211-> 1111

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition (number theory)

Column k=2 of A327639.

Cf. A063834, A328042.

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-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..37);

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]];
a[n_] := Sum[b[n, n, i] (-1)^(2 - i) Binomial[2, i], {i, 0, 2}];
a /@ Range[0, 37] (* Jean-François Alcover, May 01 2020, after Maple *)

From Vaclav Kotesovec, May 27 2020: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161... if mod(n,4) = 0
c = 96146521894.9433... if mod(n,4) = 1
c = 96146522937.2138... if mod(n,4) = 2
c = 96146521894.8218... if mod(n,4) = 3
(End)

cp's OEIS Frontend

A327631 Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows.

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