A327795 -id:A327795 - cp's OEIS frontend

A327605 Number of parts in all twice partitions of n where both partitions are strict.

0, 1, 1, 5, 8, 15, 28, 49, 86, 156, 259, 412, 679, 1086, 1753, 2826, 4400, 6751, 10703, 16250, 24757, 38047, 57459, 85861, 129329, 192660, 286177, 424358, 624510, 915105, 1347787, 1961152, 2847145, 4144089, 5988205, 8638077, 12439833, 17837767, 25536016

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

			a(3) = 5 = 1+2+2 counting the parts in 3, 21, 2|1.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000009, A279785, A327553, A327594, A327607, A327608, A327795.

Column k=2 of A327622.

g:= proc(n, i) option remember; `if`(i*(i+1)/2 f+
       [0, f[1]])(g(n-i, min(n-i, i-1)))))
    end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (f-> f+[0, f[1]*
       h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i$2))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);

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, 2][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 03 2020, after Alois P. Heinz in A327622 *)

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.

cp's OEIS Frontend

A327605 Number of parts in all twice partitions of n where both partitions are strict.

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