A015753 -id:A015753 - cp's OEIS frontend

A025147 Number of partitions of n into distinct parts >= 2.

1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 17, 21, 25, 29, 35, 41, 48, 56, 66, 76, 89, 103, 119, 137, 159, 181, 209, 239, 273, 312, 356, 404, 460, 522, 591, 669, 757, 853, 963, 1085, 1219, 1371, 1539, 1725, 1933, 2164, 2418, 2702, 3016, 3362, 3746, 4171, 4637, 5155

Offset: 0

Clark Kimberling

From R. J. Mathar, Jul 31 2008: (Start)
These "partitions of n into distinct parts >= k" and "partitions of n into distinct parts, the least being k-1" come in pairs of similar, almost shifted but not identical, sequences:
A025147, A096765 (k=2)
A025148, A096749 (k=3)
A025149, A026824 (k=4)
A025150, A026825 (k=5)
A025151, A026826 (k=6)
A025152, A026827 (k=7)
A025153, A026828 (k=8)
A025154, A026829 (k=9)
A025155, A026830 (k=10)
A096740, A026831 (k=11)
The distinction in the definitions is that "distinct parts >= k" sets a lower bound to all parts, whereas "the least being ..." means that the lower limit must be attained by one of the parts. (End)
From N. J. A. Sloane, Sep 28 2008: (Start)
Generating functions and Maple programs for the sequences in the first and second columns of the above list are respectively:
For A025147, A025148, etc.:
f:=proc(k) product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end;
For A096765, A096749, etc.:
g:=proc(k) x^(k-1)*product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end; (End)
Also number of partitions of n+1 into distinct parts, the least being 1.
Number of different sums from 1+[1,3]+[1,4]+...+[1,n]. - Jon Perry, Jan 01 2004
Also number of partitions of n such that if k is the largest part, then all parts from 1 to k occur, k occurring at least twice. Example: a(7)=3 because we have [2,2,2,1],[2,2,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n+1 such that if k is the largest part, then all parts from 1 to k occur, k occurring exactly once. Example: a(7)=3 because we have [3,2,2,1],[3,2,1,1,1] and [2,1,1,1,1,1,1] (there is a simple bijection with the partitions defined before). - Emeric Deutsch, Apr 09 2006
Also number of partitions of n+1 into distinct parts where the number of parts is itself a part. - Reinhard Zumkeller, Nov 04 2007
Partial sums give A038348 (observed by Jonathan Vos Post, proved by several correspondents).
Trivially, number of partitions of n into distinct parts (as ascending lists) such that the first part is not 1, the second not 2, the third not 3, etc., see example. - Joerg Arndt, Jun 10 2013
Convolution with A033999 gives A270144 (apart from the offset). - R. J. Mathar, Jun 18 2016

			a(7) = 3, from {{3, 4}, {2, 5}, {7}}
From _Joerg Arndt_, Jun 10 2013: (Start)
There are a(17) = 21 partitions of 17 into distinct parts >=2:
01:  [ 2 3 4 8 ]
02:  [ 2 3 5 7 ]
03:  [ 2 3 12 ]
04:  [ 2 4 5 6 ]
05:  [ 2 4 11 ]
06:  [ 2 5 10 ]
07:  [ 2 6 9 ]
08:  [ 2 7 8 ]
09:  [ 2 15 ]
10:  [ 3 4 10 ]
11:  [ 3 5 9 ]
12:  [ 3 6 8 ]
13:  [ 3 14 ]
14:  [ 4 5 8 ]
15:  [ 4 6 7 ]
16:  [ 4 13 ]
17:  [ 5 12 ]
18:  [ 6 11 ]
19:  [ 7 10 ]
20:  [ 8 9 ]
21:  [ 17 ]
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..100 from Reinhard Zumkeller)
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 798
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_1(n), see Corollary 3.7.

Cf. A015744, A015745, A015746, A015750, A015753, A015754, A015755, A002865.

a025147 = p 2 where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k + 1) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Dec 28 2011

g:=product(1+x^j,j=2..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),n=0..57); # Emeric Deutsch, Apr 09 2006
with(combstruct):ZL := {L = PowerSet(Sequence(Z,card>=2)) },unlabeled:seq(count([L,ZL],size=i),i=0..57); # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[Product[1+q^n, {n, 2, 60}], {q, 0, 60}], q]
FoldList[ PartitionsQ[ #2+1 ]-#1&, 0, Range[ 64 ] ]
(* also *)
d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 2 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 2 *)
Table[Length[d[n]], {n, 40}] (* A025147 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
p[, 0] = 1; p[k, m_] := p[k, m] = If[m < k, 0, p[k+1, m-k] + p[k+1, m]]; Table[p[2, m], {m, 0, 59}] (* Jean-François Alcover, Apr 17 2014, after Reinhard Zumkeller *)

a(n)=if(n,my(v=partitions(n));sum(i=1,#v,v[i][1]>1&&v[i]==vecsort(v[i],,8)),1) \\ Charles R Greathouse IV, Nov 20 2012

G.f.: Product_{k>=2} (1+x^k).
a(n) = A000009(n)-a(n-1) = Sum_{0<=k<=n} (-1)^k*A000009(n-k). - Henry Bottomley, May 09 2002
a(n)=t(n, 1), where t(n, k)=1+Sum_{i>j>k and i+j=n} t(i, j), 2<=k<=n. - Reinhard Zumkeller, Jan 01 2003
G.f.: 1 + Sum_{k=1..infinity} (x^(k*(k+3)/2) / Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 09 2006
The previous g.f. is a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=0} ( x^(n*(n+2*L-1)/2) / Product_{k=1..n} (1-x^k) ). - Joerg Arndt, Mar 24 2011
G.f.: Sum_{n>=1} ( x^(n*(n+1)/2-1) / Product_{k=1..n-1} (1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=L-1} ( x^(n*(n+1)/2-L*(L-1)/2) / Product_{k=1..n-(L-1)} (1-x^k) ). - Joerg Arndt, Mar 27 2011
a(n) = Sum_{1A060016(n-k+1,k-1), for n>0. - Reinhard Zumkeller, Nov 04 2007
a(n) = A096765(n+1). - R. J. Mathar, Jul 31 2008
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ 1/2 * A000009(n).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)).
(End)

Corrected and extended by Dean Hickerson, Oct 10 2001

A015744 Number of partitions of n into distinct parts, none being 2.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 22, 27, 32, 37, 44, 52, 60, 70, 82, 95, 110, 127, 146, 169, 194, 221, 254, 291, 331, 377, 429, 487, 553, 626, 707, 800, 903, 1016, 1145, 1288, 1445, 1622, 1819, 2036, 2278, 2546, 2842, 3172, 3536, 3936, 4381

Offset: 0

Clark Kimberling

nonn

With offset 2 (and a(0)=a(1)=0) the number of 2's in all partitions of n into distinct parts. [Joerg Arndt, Feb 20 2014]

			a(8)=4 because we have [8],[7,1],[5,3] and [4,3,1].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Cristina Ballantine, Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.

Cf. A025147, A015745, A015746, A015750, A015753, A015754, A015755.

g:=(1+x)*product(1+x^j,j=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..57); # Emeric Deutsch, Apr 09 2006

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^2), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 2]], {n, 0, 57}] (* Robert Price, May 17 2020 *)

G.f.: (1+x)*product(j>=3, 1+x^j ). - Emeric Deutsch, Apr 09 2006
a(n+2)=sum_{k=1..floor(n/2)} (-1)^(k-1)*A000009(n-2*k). - Mircea Merca, Feb 20 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

Corrected and extended by Dean Hickerson, Oct 10 2001

A015745 Number of partitions of n into distinct parts, none being 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 40, 47, 55, 64, 75, 87, 101, 117, 135, 155, 179, 205, 235, 269, 307, 350, 399, 453, 514, 583, 660, 746, 843, 950, 1070, 1205, 1354, 1520, 1705, 1910, 2138, 2392, 2672, 2982, 3326, 3706, 4126

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025147, A015744, A015746, A015750, A015753, A015754, A015755.

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^3), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 3]], {n, 0, 56}] (* Robert Price, May 17 2020 *)

a(n) = A015737(n+3). - Michel Marcus, Feb 20 2014

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

Corrected and extended by Dean Hickerson, Oct 10 2001

A015746 Number of partitions of n into distinct parts, none being 4.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 5, 6, 7, 9, 10, 12, 15, 18, 22, 26, 31, 36, 42, 50, 58, 68, 80, 92, 107, 124, 142, 164, 189, 216, 248, 284, 323, 369, 420, 476, 541, 613, 693, 784, 885, 997, 1123, 1264, 1419, 1593, 1787, 2000, 2239, 2504, 2795, 3120, 3479, 3874, 4313

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025147, A015744, A015745, A015750, A015753, A015754, A015755.

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^4), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 4]], {n, 0, 56}] (* Robert Price, May 17 2020 *)

a(n) = A015739(n+4). - Michel Marcus, Feb 20 2014

Corrected and extended by Dean Hickerson, Oct 10 2001

A015750 Number of partitions of n into distinct parts, none being 5.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 53, 62, 72, 84, 97, 112, 130, 150, 172, 199, 228, 260, 298, 340, 386, 440, 500, 566, 642, 727, 820, 926, 1044, 1174, 1321, 1484, 1664, 1866, 2090, 2337, 2613, 2918, 3254, 3628, 4041, 4495

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025147, A015744, A015745, A015746, A015753, A015754, A015755.

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^5), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 5]], {n, 0, 56}] (* Robert Price, May 17 2020 *)

a(n) = A015740(n+5). - Michel Marcus, Feb 20 2014

Corrected and extended by Dean Hickerson, Oct 10 2001

A015754 Number of partitions of n into distinct parts, none being 7.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 10, 12, 14, 18, 22, 25, 30, 36, 42, 50, 58, 67, 79, 92, 106, 123, 142, 164, 189, 217, 248, 284, 325, 370, 421, 479, 543, 616, 698, 788, 890, 1005, 1131, 1273, 1432, 1606, 1802, 2020, 2259, 2527, 2824, 3150, 3514, 3916, 4358, 4849

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025147, A015744, A015745, A015746, A015750, A015753, A015755.

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^7), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 7]], {n, 0, 56}] (* Robert Price, May 18 2020 *)

a(n) = A015742(n+7). - Michel Marcus, Feb 20 2014

Corrected and extended by Dean Hickerson, Oct 10 2001

A015755 Number of partitions of n into distinct parts, none being 8.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 13, 15, 18, 22, 27, 31, 37, 44, 51, 61, 71, 82, 95, 111, 128, 148, 171, 195, 225, 258, 295, 337, 384, 437, 497, 565, 639, 724, 818, 923, 1042, 1173, 1319, 1483, 1665, 1866, 2092, 2341, 2616, 2924, 3263, 3637, 4053, 4512, 5016

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025147, A015744, A015745, A015746, A015750, A015753, A015754.

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^8), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 8]], {n, 0, 56}] (* Robert Price, May 18 2020 *)

a(n) = A015743(n+8). - Michel Marcus, Feb 20 2014

a(n) = A000009(n) - a(n-8) for n >= 0, a(n) = 0 otherwise. - Yuchun Ji, Dec 25 2018

Corrected and extended by Dean Hickerson, Oct 10 2001

A015741 Number of 6's in all the partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, 17, 21, 24, 29, 34, 40, 47, 55, 65, 75, 88, 102, 118, 137, 157, 181, 208, 238, 272, 311, 355, 404, 460, 522, 592, 671, 758, 856, 966, 1088, 1224, 1377, 1546, 1734, 1944

Offset: 1

Clark Kimberling

nonn

a(n+6) = A015753(n). - Alois P. Heinz, Aug 24 2011

			a(9) = 2 because in the 8 (=A000009(9)) partitions of 9 into distinct parts, namely [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2] we have altogether two parts equal to 6.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A015753.

g:=x^6*product(1+x^j,j=1..60)/(1+x^6): gser:=series(g,x=0,57): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 17 2006

nmax = 100; Rest[CoefficientList[Series[x^6/(1+x^6) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
Table[Count[Flatten@Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], 6], {n, 54}] (* Robert Price, May 16 2020 *)

G.f.: x^6 * product(j>=1, 1+x^j )/(1+x^6). - Emeric Deutsch, Apr 17 2006
Corresponding g.f. for "number of k's" is x^k/(1+x^k)*prod(n>=1, 1+x^n ). [Joerg Arndt, Feb 20 2014]
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

cp's OEIS Frontend

A025147 Number of partitions of n into distinct parts >= 2.

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A015744 Number of partitions of n into distinct parts, none being 2.

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A015745 Number of partitions of n into distinct parts, none being 3.

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A015746 Number of partitions of n into distinct parts, none being 4.

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A015750 Number of partitions of n into distinct parts, none being 5.

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A015754 Number of partitions of n into distinct parts, none being 7.

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A015755 Number of partitions of n into distinct parts, none being 8.

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