A024791 -id:A024791 - cp's OEIS frontend

A024786 Number of 2's in all partitions of n.

0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by Emeric Deutsch, Aug 13 2008]
In general the number of times that j appears in the partitions of n equals Sum_{kA024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
Equals row sums of triangle A173238. - Gary W. Adamson, Feb 13 2010
The sums of two successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - Emeric Deutsch, Sep 12 2016
a(n) is also the number of non-isomorphic vertex-transitive cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n-1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019
Assuming a partition is in weakly decreasing order, a(n) is also the number of times -1 occurs in the differences of the partitions of n+1. - George Beck, Mar 28 2023

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
.                             Number
Partitions of 7               of 2's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 0
5 + 2 .......................... 1
3 + 2 + 2 ...................... 2
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 3
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 1
4 + 1 + 1 + 1 .................. 0
2 + 2 + 1 + 1 + 1 .............. 2
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 1
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
.  24 - 13 =                    11
.
The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.
(End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Cuthbertson, Fixed hooks in arbitrary columns of partitions, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Emeric Deutsch et al., Problem 11237, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From _Emeric Deutsch_, Aug 13 2008]
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.

Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.
Column 2 of A060244.
First differences of A000097.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, May 18 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)
Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *)
Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 27 2017 *)

from sympy import npartitions
def A024786(n): return sum(npartitions(n-(k<<1)) for k in range(1,(n>>1)+1)) # Chai Wah Wu, Oct 25 2023

a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - Christian G. Bower, Jun 22 2000
a(n) = Sum_{kA000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
G.f.: (x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.
a(n) = A006128(n-1) - A194452(n-1). - Omar E. Pol, Nov 20 2011
a(n) = A181187(n,2) - A181187(n,3). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - Vaclav Kotesovec, Mar 07 2016, extended Nov 05 2016
a(n) = Sum_{k} k * A116595(n-1,k). - Emeric Deutsch, Sep 12 2016
G.f.: x^2/((1 - x)*(1 - x^2)) * Sum_{n >= 0} x^(2*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A004526 (partitions into 2 parts, or, modulo offset differences, partitions into parts <= 2) and A002865 (partitions into parts >= 2). - Peter Bala, Jan 17 2021

A024788 Number of 4's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 105, 139, 190, 250, 336, 436, 575, 740, 963, 1228, 1577, 1995, 2538, 3186, 4013, 5005, 6256, 7751, 9617, 11847, 14605, 17894, 21927, 26730, 32582, 39531, 47942, 57915, 69920, 84114, 101116, 121176, 145095, 173248

Offset: 1

Clark Kimberling

The sums of four successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 4th largest and the sum of 5th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(n+4) is the number of n-vertex graphs that do not contain a triangle, P_4, or K_2,3 as induced subgraph. These are the K_2,3-free bipartite cographs. Bipartite cographs are graph that are disjoint unions of complete bipartite graphs [Babel et al. Corollary 2.2], and forbidding K_2,3 leaves one possible component for each size except size 4, where there are two. Thus, this number is A000041(n) + a(n) = a(n+4). - Falk Hüffner, Jan 11 2016
a(n) (n>=3) is the number of even singletons in all partitions of n-2 (by a singleton we mean a part that occurs exactly once). Example: a(7) = 3 because in the partitions [5], [4*,1], [3,2*], [3,1,1], [2,2,1], [2*,1,1,1], [1,1,1,1,1] we have 3 even singletons (marked by *). The statement of this comment can be obtained by setting k=2 in Theorem 2 of the Andrews et al. reference. - Emeric Deutsch, Sep 13 2016

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
.                             Number
Partitions of 7               of 4's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 0
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 0
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 0
4 + 1 + 1 + 1 .................. 1
2 + 2 + 1 + 1 + 1 .............. 0
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
.           7 - 4 =              3
The difference between the sum of the fourth column and the sum of the fifth column of the set of partitions of 7 is 7 - 4 = 3 and equals the number of 4's in all partitions of 7, so a(7) = 3.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews and E. Deutsch, A note on a method of Erdos and the Stanley-Elder theorems, Integers, 16 (2016), A24.
L. Babel, A. Brandstädt, and V. B. Le, Recognizing the P4-structure of bipartite graphs, Discrete Appl. Math. 93 (1999), 157-168.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A000041, A066633, A024786, A024787, A024789, A024790, A024791, A024792, A024793, A024794.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i=4, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 4], {n, 1, 50} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 4, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,4) - A181187(n,5). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+4) - a(n) = A000041(n). a(n) + a(n+2) = A024786(n).
O.g.f.: x^4/(1 - x^4) * product {k >= 1} 1/(1 - x^k) = x^4 + x^5 + 2*x^6 + 3*x^7 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)) * (1 - 49*Pi/(24*sqrt(6*n)) + (49/48 + 1633*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^4/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)) * Sum_{n >= 0} x^(4*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A026810 (partitions into 4 parts, or, modulo offset differences, partitions into parts <= 4) and A008484 (partitions into parts >= 4). - Peter Bala, Jan 17 2021

A024787 Number of 3's in all partitions of n.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, 164, 222, 298, 395, 519, 683, 885, 1146, 1475, 1887, 2401, 3050, 3845, 4837, 6060, 7563, 9402, 11664, 14405, 17751, 21807, 26715, 32634, 39784, 48352, 58649, 70969, 85690, 103232, 124143, 148951, 178407, 213277, 254509

Offset: 1

Clark Kimberling

Starting with the first 1 = row sums of triangle A173239. - Gary W. Adamson, Feb 13 2010
The sums of three successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 3rd largest and the sum of 4th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
.                             Number
Partitions of 7               of 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
6 + 1 .......................... 0
3 + 3 + 1 ...................... 2
4 + 2 + 1 ...................... 0
2 + 2 + 2 + 1 .................. 0
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 1
4 + 1 + 1 + 1 .................. 0
2 + 2 + 1 + 1 + 1 .............. 0
3 + 1 + 1 + 1 + 1 .............. 1
2 + 1 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
.      13 - 7 =                  6
The difference between the sum of the third column and the sum of the fourth column of the set of partitions of 7 is 13 - 7 = 6 and equals the number of 3's in all partitions of 7, so a(7) = 6.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..5000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Cuthbertson, Fixed hooks in arbitrary columns of partitions, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A066633, A024786, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A263232.

Cf. A173239. - Gary W. Adamson, Feb 13 2010

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=3, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 3], {n, 1, 50} ]
b[n_, i_] := b[n, i] = Module[{g}, If[n==0 || i==1, {1, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[i==3, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)
Join[{0, 0}, (1/((1 - x^3) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *)

a(n) = A181187(n,3) - A181187(n,4). - Omar E. Pol, Oct 25 2012
a(n) = Sum_{k=1..floor(n/3)} A263232(n,k). - Alois P. Heinz, Nov 01 2015
a(n) ~ exp(Pi*sqrt(2*n/3)) / (6*Pi*sqrt(2*n)) * (1 - 37*Pi/(24*sqrt(6*n)) + (37/48 + 937*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^2/((1 - x^3)*(x)inf), where (q)_inf is the q-Pochhammer symbol (the Euler function). - _Vladimir Reshetnikov, Nov 22 2016
G.f.: x^3/((1 - x)*(1 - x^2)*(1 - x^3)) * Sum_{n >= 0} x^(3*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A069905 (partitions into 3 parts, or, modulo offset differences, partitions into parts <= 3) and A008483 (partitions into parts >= 3). - Peter Bala, Jan 17 2021

A024789 Number of 5's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 68, 94, 126, 170, 226, 299, 391, 511, 660, 853, 1091, 1393, 1766, 2235, 2811, 3527, 4403, 5484, 6800, 8415, 10369, 12752, 15627, 19110, 23298, 28346, 34389, 41642, 50295, 60636, 72929, 87563, 104903, 125470

Offset: 1

Clark Kimberling

nonn

The sums of five successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 5th largest and the sum of 6th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 8 we have:
--------------------------------------
.                             Number
Partitions of 8               of 5's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 0
5 + 3 .......................... 1
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 0
2 + 2 + 2 + 2 .................. 0
7 + 1 .......................... 0
4 + 3 + 1 ...................... 0
5 + 2 + 1 ...................... 1
3 + 2 + 2 + 1 .................. 0
6 + 1 + 1 ...................... 0
3 + 3 + 1 + 1 .................. 0
4 + 2 + 1 + 1 .................. 0
2 + 2 + 2 + 1 + 1 .............. 0
5 + 1 + 1 + 1 .................. 1
3 + 2 + 1 + 1 + 1 .............. 0
4 + 1 + 1 + 1 + 1 .............. 0
2 + 2 + 1 + 1 + 1 + 1 .......... 0
3 + 1 + 1 + 1 + 1 + 1 .......... 0
2 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .. 0
------------------------------------
.               7 - 4 =          3
The difference between the sum of the fifth column and the sum of the sixth column of the set of partitions of 8 is 7 - 4 = 3 and equals the number of 5's in all partitions of 8, so a(8) = 3.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A066633, A024786, A024787, A024788, A024790, A024791, A024792, A024793, A024794.

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=5, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 5], {n, 1, 50} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 5, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

x='x+O('x^50); concat([0, 0, 0, 0], Vec(x^5/(1 - x^5) * prod(k=1, 50, 1/(1 - x^k)))) \\ Indranil Ghosh, Apr 06 2017

a(n) = A181187(n,5) - A181187(n,6). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*Pi*sqrt(2*n)) * (1 - 61*Pi/(24*sqrt(6*n)) + (61/48 + 2521*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^5/(1 - x^5) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 06 2017

A024790 Number of 6's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 89, 117, 159, 209, 278, 360, 474, 607, 786, 1001, 1280, 1615, 2049, 2565, 3222, 4011, 4998, 6180, 7653, 9407, 11571, 14154, 17308, 21063, 25630, 31044, 37586, 45339, 54646, 65646, 78804, 94305, 112761, 134473

Offset: 1

Clark Kimberling

The sums of six successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 6th largest and the sum of 7th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024791, A024792, A024793, A024794.

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=6, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 6], {n, 1, 52} ]
b[n_, i_] := b[n, i] = Module[{g}, If [n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 6, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,6) - A181187(n,7). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+6) - a(n) = A000041(n). a(n) + a(n+3) = A024787(n).
a(n) + a(n+2) + a(n+4) = A024786(n).
O.g.f.: x^6/(1 - x^6) * product {k >= 1} 1/(1 - x^k) = x^6 + x^7 + 2*x^8 + 3*x^9 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*Pi*sqrt(2*n)) * (1 - 73*Pi/(24*sqrt(6*n)) + (73/48 + 3601*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

A024794 Number of 10's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 79, 104, 140, 183, 242, 312, 407, 520, 670, 849, 1081, 1359, 1715, 2141, 2678, 3322, 4125, 5085, 6274, 7691, 9430, 11502, 14025, 17024, 20655, 24959, 30140, 36270, 43612, 52274, 62604, 74763

Offset: 1

Clark Kimberling

The sums of ten successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 10th largest and the sum of 11th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
In general, if m>0 and a(n+m)-a(n) = A000041(n), then a(n) ~ exp(sqrt(2*n/3)*Pi) / (2*Pi*m*sqrt(2*n)) * (1 - Pi*(1/24 + m/2)/sqrt(6*n) + (1/48 + Pi^2/6912 + m/4 + m*Pi^2/288 + m^2*Pi^2/72)/n). - Vaclav Kotesovec, Nov 05 2016

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.

Cf. A066633, A000070(n-1), A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A000041.

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=10, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 10], {n, 1, 55} ]
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 10, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,10) - A181187(n,11). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+10) - a(n) = A000041(n). a(n) + a(n+5) = A024789(n).
a(n) + a(n+2) + a(n+4) + a(n+6) + a(n+8) = A024786(n).
O.g.f.: x^10/(1 - x^10) * product {k >= 1} 1/(1 - x^k) = x^10 + x^11 + 2*x^12 + 3*x^13 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (20*Pi*sqrt(2*n)) * (1 - 121*Pi/(24*sqrt(6*n)) + (121/48 + 9841*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

A024792 Number of 8's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 44, 59, 82, 108, 146, 191, 254, 328, 429, 549, 709, 900, 1148, 1446, 1829, 2286, 2865, 3559, 4427, 5465, 6752, 8288, 10178, 12429, 15175, 18442, 22404, 27102, 32767, 39473, 47516, 57012, 68349, 81703, 97579, 116236

Offset: 1

Clark Kimberling

The sums of eight successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 8th largest and the sum of 9th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024793, A024794.

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=8, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 8], {n, 1, 53} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 8, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,8) - A181187(n,9). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+8) - a(n) = A000041(n). a(n) + a(n+4) = A024788(n).
a(n) + a(n+2) + a(n+4) + a(n+6) = A024786(n).
O.g.f.: x^8/(1 - x^8) * product {k >= 1} 1/(1 - x^k) = x^8 + x^9 + 2*x^10 + 3*x^11 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*Pi*sqrt(2*n)) * (1 - 97*Pi/(24*sqrt(6*n)) + (97/48 + 6337*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

A024793 Number of 9's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 43, 58, 80, 106, 142, 187, 246, 319, 416, 533, 685, 872, 1108, 1397, 1762, 2204, 2755, 3426, 4251, 5250, 6476, 7950, 9746, 11905, 14514, 17638, 21403, 25888, 31265, 37661, 45288, 54329, 65079, 77775

Offset: 1

Clark Kimberling

The sums of nine successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 9th largest and the sum of 10th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024794.

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=9, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 9], {n, 1, 55} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 9, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,9) - A181187(n,10). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+9) - a(n) = A000041(n).
a(n) + a(n+3) + a(n+6) = A024787(n).
O.g.f.: x^9/(1 - x^9) * product {k >= 1} 1/(1 - x^k) = x^9 + x^10 + 2*x^11 + 3*x^12 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (18*Pi*sqrt(2*n)) * (1 - 109*Pi/(24*sqrt(6*n)) + (109/48 + 7993*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

A206557 Number of 7's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 23, 28, 39, 48, 64, 79, 104, 128, 165, 204, 258, 317, 399, 487, 606, 739, 912, 1105, 1356, 1637, 1994, 2400, 2906, 3485, 4199, 5016, 6015, 7164, 8553, 10151, 12076, 14286, 16930, 19974, 23588, 27749

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024791. Also number of occurrences of 7 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of seven successive terms give the partition numbers A000041.

Column 7 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555, A206556, A206558-A206560.

A206557 = lambda n: sum(list(p).count(7) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..7} a(n+j), n >= 0.

A179385 The n-th term is the sum of all the 1's generated from all the combinations of prime numbers and ones possible, that add to n, when each prime is only allowed once and any number of ones are allowed.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 20, 27, 35, 44, 55, 67, 81, 97, 115, 135, 158, 183, 212, 244, 280, 320, 364, 413, 467, 526, 591, 661, 737, 820, 909, 1007, 1112, 1226, 1349, 1481, 1624, 1778, 1943, 2121, 2311, 2515, 2734, 2968, 3219, 3486, 3771, 4075, 4399, 4744, 5112, 5502

Offset: 1

Joseph Foley, Jul 12 2010

nonn

			n=7 gives 11111 11, 2111 11, 311 11, 5 11, 5 2, 32 11. (Grouped in 5's) no. of 1's: 7, 5, 4, 2, 0, 2. Sum is 20, therefore a(7) = 20.
n=12 gives 11111 11111 11, 11111 11111 2, 11111 311 11, 11111 32 11, 11111 5 11, 5 2111 11, 5 311 11, 5 32 11, 7111 11, 721 11, 73 11, 73 2, 75, eleven 1, no. of 1's: 12, 10, 9, 7, 7, 5, 4, 2, 5, 3, 2, 0, 0, 1. Sum is 67, therefore a(12) = 67.
1: 1 => 1 2: 11, 2 => 2 3: 111, 21 => 4 4: 1111, 211, 22, 31 => 7 5: 11111, 2111, 311, 23 => 10 6: 11111 1, 2111 1, 311 1, 23 1, 5 1 => 15 and so on.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 175 terms from Robert G. Wilson v)

Cf. A000586.
Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
Partial sums of A280271.

b:= proc(n,i) option remember; if n<=0 then 0 elif i=0 then n else b(n, i-1) +b(n-ithprime(i), i-1) fi end: # R. J. Mathar, Jul 14 2010
a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=1..80); # Alois P. Heinz

fQ[lst_List] := Sort@ Flatten@ Most@ Split@ lst == Rest@ Union@ lst; f[n_] := Sum[ Count[ Select[ IntegerPartitions[n, {k}, Join[{1}, Prime@ Range@ PrimePi@n]], fQ@# &], 1, 2], {k, n}]; Array[f, 50] (* improved by Robert G. Wilson v, Jul 20 2010 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[Prime[i] > n, 0, b[n - Prime[i], i - 1]]]];
a[n_] := Sum[k*b[n - k, PrimePi[n - k]], {k, 1, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = my(r); r = x/(1-x)^2 + O(x^(n+1)); forprime(p=2,n,r*=1+x^p); polcoeff(r,n) \\ Max Alekseyev, Jul 14 2010

a(n) = Sum_{k=1..n} k * A000586(n-k). - Max Alekseyev, Jul 14 2010

Corrected and extended by R. J. Mathar, Jul 14 2010

cp's OEIS Frontend

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A024790 Number of 6's in all partitions of n.

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A024794 Number of 10's in all partitions of n.

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A024792 Number of 8's in all partitions of n.

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