A327605 -id:A327605 - cp's OEIS frontend

A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

1, 1, 1, 3, 4, 7, 11, 18, 28, 47, 71, 108, 166, 252, 382, 587, 869, 1282, 1938, 2832, 4153, 6148, 8962, 12965, 18913, 27301, 39380, 56747, 81226, 115907, 166358, 236000, 334647, 475517, 671806, 947552, 1335679, 1875175, 2630584, 3687589, 5150585, 7183548

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - Seiichi Manyama, Nov 14 2018

			The a(6)=11 twice-partitions are:
((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A000009, A063834, A270995, A279375, A327553, A327605.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

nn=20;CoefficientList[Series[Product[(1+PartitionsQ[k]x^k),{k,nn}],{x,0,nn}],x]
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n]; b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: Product_{k>0} (1 + A000009(k)*x^k). - Seiichi Manyama, Nov 14 2018

A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; 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, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 3, 1, 0, 1, 1, 7, 8, 5, 1, 0, 1, 1, 9, 16, 15, 8, 1, 0, 1, 1, 11, 27, 35, 28, 10, 1, 0, 1, 1, 13, 41, 69, 73, 49, 13, 1, 0, 1, 1, 15, 58, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327632.

			Square array A(n,k) begins:
  0,  0,  0,   0,    0,    0,    0,     0,     0, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  3,  5,   7,    9,   11,   13,    15,    17, ...
  1,  3,  8,  16,   27,   41,   58,    78,   101, ...
  1,  5, 15,  35,   69,  121,  195,   295,   425, ...
  1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
  1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
  1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A057427, A015723, A327605, A327628.
Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.
Main diagonal gives: A327623.
Cf. A327618, A327632.

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:
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] = 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_, k_] := b[n, n, k][[2]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).

A327594 Number of parts in all twice partitions of n.

Original entry on oeis.org

0, 1, 5, 14, 44, 100, 274, 581, 1417, 2978, 6660, 13510, 29479, 58087, 120478, 236850, 476913, 916940, 1812498, 3437043, 6657656, 12512273, 23780682, 44194499, 83117200, 152837210, 283431014, 517571202, 949844843, 1719175176, 3127751062, 5618969956, 10133425489

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

			a(2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
a(3) = 14 = 1+2+3+2+3+3: 3, 21, 111, 2|1, 11|1, 1|1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..3200

Cf. A000041, A006128, A063834, A327590, A327605, A327607, A327608.

Column k=2 of A327618.

g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
      numtheory[tau](j), j=1..n)])(combinat[numbpart])
    end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
       h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
# second Maple program:
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-> b(n$2, 2)[2]:
seq(a(n), n=0..37);

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_] := b[n, n, 2][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

A327607 Number of parts in all twice partitions of n where the first partition is strict.

Original entry on oeis.org

0, 1, 3, 11, 21, 58, 128, 276, 516, 1169, 2227, 4324, 8335, 15574, 29116, 55048, 97698, 176291, 323277, 563453, 1005089, 1770789, 3076868, 5293907, 9184885, 15668638, 26751095, 45517048, 76882920, 128738414, 217219751, 360525590, 599158211, 995474365

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

			a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A000009, A000041, A271619, A327552, A327594, A327605, A327608.

g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
      numtheory[tau](j), j=1..n)])(combinat[numbpart])
    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))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);

g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

A327608 Number of parts in all twice partitions of n where the second partition is strict.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 74, 134, 254, 470, 842, 1463, 2620, 4416, 7545, 12749, 21244, 34913, 57868, 93583, 151963, 244602, 391206, 620888, 987344, 1550754, 2435087, 3804354, 5920225, 9162852, 14179754, 21785387, 33436490, 51121430, 77935525, 118384318, 179617794

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

			a(3) = 8 = 1+2+2+3 counting the parts in 3, 21, 2|1, 1|1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A000009, A000041, A270995, A327554, A327594, A327605, A327607.

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`(n=0, [1, 0],
      `if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
       h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i$2)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);

g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, g[n, i - 1] + Function[f, f + {0, f[[1]]}][g[n - i, Min[n - i, i - 1]]]]];
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 2, 0, b[n, i - 1]] + Module[{h, f}, h = g[i, i]; f = b[n - i, Min[n - i, i]] h[[1]]; f + {0, f[[1]] h[[2]]/h[[1]]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

A327553 Number of partitions in all twice partitions of n where both partitions are strict.

Original entry on oeis.org

0, 1, 1, 4, 6, 11, 20, 33, 57, 100, 165, 254, 417, 649, 1039, 1648, 2540, 3836, 6020, 9035, 13645, 20752, 31054, 45993, 68668, 101511, 149525, 220132, 321614, 468031, 684124, 989703, 1427054, 2064859, 2964987, 4254028, 6090453, 8686574, 12366583, 17598885

Offset: 0

Alois P. Heinz, Sep 16 2019

nonn

			a(3) = 4 = 1+1+2 counting the partitions in 3, 21, 2|1.

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

Cf. A000009, A279785, A327605.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 p+[0, p[1]])(
       g(i)*b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, {0, 0}, If[n==0, {1, 0}, b[n, i-1] + Function[p, p + {0, p[[1]]}][g[i] b[n-i, Min[n-i, i-1]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

A327795 Number of parts in all proper twice partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 0, 3, 6, 13, 30, 61, 121, 210, 353, 600, 989, 1628, 2667, 4205, 6514, 10406, 15893, 24322, 37516, 56824, 85102, 128420, 191579, 284898, 422839, 622721, 913006, 1345320, 1958269, 2843788, 4140170, 5983662, 8632808, 12433730, 17830728, 25527909, 36516161

Offset: 1

Alois P. Heinz, Sep 25 2019

nonn

			a(4) = 3:
  4 -> 31 -> 211   (3 parts)

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Column k=2 of A327632.

Cf. A327605.

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:
a:= n-> (k-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=1..41);

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}];
a[n_] := T[n, 2];
Array[a, 41] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

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A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

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A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A327594 Number of parts in all twice partitions of n.

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Maple

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A327607 Number of parts in all twice partitions of n where the first partition is strict.

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Maple

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A327608 Number of parts in all twice partitions of n where the second partition is strict.

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Maple

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A327553 Number of partitions in all twice partitions of n where both partitions are strict.

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Maple

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A327795 Number of parts in all proper twice partitions of n into distinct parts.

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Maple

Mathematica