A327608 -id:A327608 - cp's OEIS frontend

A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909

Offset: 0

Vaclav Kotesovec, Mar 28 2016

nonn

The number of ways a number can be partitioned into not necessarily distinct parts and then each part is partitioned into distinct parts. Also a(n) > A089259(n) for n>5. - Gus Wiseman, Apr 10 2016
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers with weakly decreasing multiplicities. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 7 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{2},{1,1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The weakly normal non-strict version is A055887.
The non-strict version is A063834.
The weakly normal version is A304969.
(End)

			a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For compositions instead of partitions we have A304969, non-strict A055887.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.

nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

From Vaclav Kotesovec, Mar 28 2016: (Start)
a(n) ~ c * n^2 * 2^(n/3), where
c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0
c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1
c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2
(End)

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

Original entry on oeis.org

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 *)

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 *)

A327554 Number of partitions in all twice partitions of n where the second partition is strict.

Original entry on oeis.org

0, 1, 3, 7, 15, 29, 60, 108, 201, 364, 643, 1106, 1944, 3253, 5493, 9183, 15161, 24727, 40559, 65173, 104963, 167747, 266452, 420329, 663658, 1036765, 1618221, 2514169, 3891121, 5992868, 9224213, 14107699, 21548428, 32798065, 49779331, 75301296, 113757367

Offset: 0

Alois P. Heinz, Sep 16 2019

nonn

			a(3) = 7 = 1+1+2+3 counting the partitions in 3, 21, 2|1, 1|1|1.

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

Cf. A000009, A000041, A270995, A327608.

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

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A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

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

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A327594 Number of parts in all twice partitions of n.

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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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A327554 Number of partitions in all twice partitions of n where the second partition is strict.

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Maple

Mathematica