A000009 -id:A000009 - cp's OEIS frontend

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004
Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018
Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

Offset: 0

N. J. A. Sloane, Dec 23 2003

nonn

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

			From _Gus Wiseman_, Oct 22 2018: (Start)
The a(6) = 22 set multipartitions of integer partitions of 6:
  (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
       (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
       (1)(5)  (1)(23)    (1)(2)(12)
       (2)(4)  (2)(13)    (1)(1)(1)(3)
       (3)(3)  (3)(12)    (1)(1)(2)(2)
               (1)(1)(4)
               (1)(2)(3)
               (2)(2)(2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
      if n<0 or n>i*(i+1)/2 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n,i):= b(n-i, i-1) +b(n, i-1)
      fi
    end:
a:= proc(n) option remember; `if` (n=0, 1,
       add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 11 2011

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

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

Original entry on oeis.org

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)

A036469 Partial sums of A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88, 110, 137, 169, 207, 253, 307, 371, 447, 536, 640, 762, 904, 1069, 1261, 1483, 1739, 2035, 2375, 2765, 3213, 3725, 4310, 4978, 5738, 6602, 7584, 8697, 9957, 11383, 12993, 14809, 16857, 19161, 21751, 24661

Offset: 0

N. J. A. Sloane

nonn

Also number of 1's in all partitions of n+1 into odd parts. Example: a(4)=7 because the partitions of 5 into odd parts are [5], [3,1,1], [1,1,1,1,1], having a total number of 7 1's. - Emeric Deutsch, Mar 29 2006
Convolved with A035363 = A000070. - Gary W. Adamson, Jun 09 2009
Equals row sums of triangle A166240. - Gary W. Adamson, Oct 09 2009
a(n) = if n <= 1 then A201377(1,n) else A201377(n,1). - Reinhard Zumkeller, Dec 02 2011
a(n) equals the sum of the parts of the form 2^k (k >= 0) in all partitions of n + 1 into distinct parts. Example: a(6) = 14. The partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] having sum over parts of the form 2^k equal to 1 + 2 + 4 + 4 + 2 + 1 = 14. - Peter Bala, Dec 01 2013
Number of partitions of the (n+1)-multiset {0,...,0,1} with n 0's into distinct multisets; a(3) = 5: 0|00|1, 00|01, 000|1, 0|001, 0001.  Also number of factorizations of 3*2^n into distinct factors; a(3) = 5: 2*3*4, 4*6, 3*8, 2*12, 24. - Alois P. Heinz, Jul 30 2021

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
A. V. Chekhonadskikh, Some classical number sequences in control system design, Siberian Electronic Mathematical Reports, Volume 14, p. 620-628. See Theorem 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 774

Cf. A000009, A265093.
Cf. A035363, A000070. - Gary W. Adamson, Jun 09 2009
Cf. A166240. - Gary W. Adamson, Oct 09 2009
Column k=1 of A346520.

g:=1/(1-x)/product(1-x^(2*j-1),j=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..46); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

CoefficientList[ Series[Product[(1 + t^i), {i, 1, Infinity}]/(1 - t), {t, 0, 46}], t] (* Geoffrey Critzer, May 16 2010 *)
b[n_, i_] := If[n == 0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, Min[n-i, i-1]]]]]; a[n_] := a[n] = b[n, n]+If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
Accumulate[Table[PartitionsQ[n], {n, 0, 50}]] (* Vaclav Kotesovec, Oct 26 2016 *)

G.f.: 1/[(1-x)*product(1-x^(2j-1), j=1..infinity)]. - Emeric Deutsch, Mar 29 2006
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)) * (1 + (18+13*Pi^2) / (48*Pi*sqrt(3*n)) + (2916 - 1404*Pi^2 + 121*Pi^4)/(13824*Pi^2*n)). - Vaclav Kotesovec, Feb 26 2015, updated Oct 26 2016
For n > 0, a(n) = A026906(n) + 1. - Vaclav Kotesovec, Oct 26 2016
Faster converging g.f.: A(x) = (1/(1 - x))*Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). - Peter Bala, Feb 02 2021

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

			4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
From _Gus Wiseman_, Oct 11 2018: (Start)
a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
  {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}
                {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}
                {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}
                           {{1},{1,2}}  {{1},{4}}    {{1,2,3}}
                                        {{2},{3}}    {{1},{5}}
                                        {{1},{1,3}}  {{2},{4}}
                                        {{2},{1,2}}  {{1},{1,4}}
                                                     {{1},{2,3}}
                                                     {{2},{1,3}}
                                                     {{3},{1,2}}
                                                     {{1},{2},{3}}
                                                     {{1},{2},{1,2}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000
N. J. A. Sloane, Transforms

Cf. A050343-A050350, A089254.
Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
Row sums of A330462 and of A360764.

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)
nn=10;Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k],{k,nn}],{x,0,n}],{n,0,nn}] (* Gus Wiseman, Oct 11 2018 *)

Weigh transform of A000009.

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A000009.
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. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,1}}    {{1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The non-strict version is A055887.
The strongly normal non-strict version is A063834.
The strongly normal version is A270995.
(End)

			From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
  ((1))  ((2))     ((3))        ((4))
         ((1)(1))  ((21))       ((31))
                   ((1)(2))     ((1)(3))
                   ((2)(1))     ((2)(2))
                   ((1)(1)(1))  ((3)(1))
                                ((1)(21))
                                ((21)(1))
                                ((1)(1)(2))
                                ((1)(2)(1))
                                ((2)(1)(1))
                                ((1)(1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2816
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions
Index entries for sequences related to compositions

Cf. A050342, A055887, A067687, A081362, A279785, A299106.
Row sums of A308680.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For partitions instead of compositions we have A270995, non-strict A063834.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.
Cf. A279784.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

Offset: 0

Alois P. Heinz, Sep 05 2014

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

			The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3;
   5,  6;
   7, 10;
  11, 14, 15;
  13, 21, 22, 30;
  17, 26, 33, 35, 42;
  19, 34, 39, 55, 66,  70;
  23, 38, 51, 65, 77,  78, 105, 110;
  29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
  ...
Corresponding triangle of strict integer partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (42) (51) (321)
        (7) (61) (52) (43) (421)
     (8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018

Alois P. Heinz, Rows n = 0..42, flattened

Column k=1 gives: A008578(n+1).
Last elements of rows give: A246868.
Row sums give A147655.
Row lengths are: A000009.
Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
      map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 0, 0, -1, -3, -7, -14, -25, -41, -64, -100, -165, -294, -550, -1023, -1795, -2823, -3658, -2882, 2873, 20435, 62185, 148863, 314008, 613957, 1155794, 2175823, 4244026, 8753538, 19006490, 42471787, 95234575, 210395407, 453413866, 949508390, 1931939460

Offset: 0

Vaclav Kotesovec, Oct 09 2017

sign

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph: a(n)/2^n (40000 terms)

Cf. A025147, A266232, A294467, A294468.
Cf. A218481, A294466, A095051.
The non-strict version is the absolute value of A281425; see A175804, A320590.
Up to sign, same as A380412. See A320591, A377285, A378970, A378971.
A000009 counts strict integer partitions, differences A087897.
Cf. A000041, A002865, A007442, A008284, A116608.

Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

Original entry on oeis.org

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852

Offset: 1

Olivier Gérard

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

			From _Gus Wiseman_, Jan 13 2020: (Start)
Strict partitions of a(1) = 3 through a(4) = 7:
  (3)    (4)    (5)    (7)
  (2,1)  (3,1)  (3,2)  (4,3)
                (4,1)  (5,2)
                       (6,1)
                       (4,2,1)
(End)

Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Partition Function Q
Eric Weisstein's World of Mathematics, Partition Function Q-Congruences

The non-strict version is A046063.
The version for powers of 2 instead of primes is A331022.
The version for factorizations instead of strict partitions is A330991.
The version for strict factorizations instead of strict partitions is A331201.
Cf. A000009, A051005, A056848, A265835.

n = 1; A035359 = {}; While[n < 10^7, n++; If[ PrimeQ[ PartitionsQ[n]], Print[n]; AppendTo[A035359, n]]]; A035359 (* Jean-François Alcover, Oct 12 2011 *)

More terms from Eric W. Weisstein
a(12) from Max Alekseyev, Jul 04 2009
a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015
a(15)-a(17) from Max Alekseyev, Jul 10 2015

cp's OEIS Frontend

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

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A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

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

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A036469 Partial sums of A000009 (partitions into distinct parts).

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A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

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A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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