A066898 -id:A066898 - cp's OEIS frontend

A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n.

0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637

Offset: 0

N. J. A. Sloane, Colin Mallows

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989).) - Wolfdieter Lang
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g., for n=3 to n=2 we have a(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, see A066186. - Thomas Wieder, May 20 2004
Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel, Nov 07 2005
Starting with offset 1, = the partition triangle A008284 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008
Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11, ...) convolved with A000005: (1, 2, 2, 3, 2, 4, ...). - Gary W. Adamson, Jun 16 2009
Apart from initial 0 row sums of triangle A066633, also the Möbius transform is A085410. - Gary W. Adamson, Mar 21 2011
More generally, the total number of parts >= k in all partitions of n equals the sum of k-th largest parts of all partitions of n. In this case k = 1. Apart from initial 0 the first column of A181187. - Omar E. Pol, Feb 14 2012
Row sums of triangle A221530. - Omar E. Pol, Jan 21 2013
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the total number of divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also as follows:
Convolution of A000005 and A000041.
Convolution of A006218 and A002865.
Convolution of A341062 and A000070.
Row sums of triangles A221531, A245095, A339258, A340525, A340529. (End)
Number of ways to choose a part index of an integer partition of n, i.e., partitions of n with a selected position. Selecting a part value instead of index gives A000070. - Gus Wiseman, Apr 19 2021

			For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. The total number of parts is 12. On the other hand, the sum of the largest parts of all partitions is 4 + 2 + 3 + 2 + 1 = 12, equaling the total number of parts, so a(4) = 12. - _Omar E. Pol_, Oct 12 2018

S. M. Luthra, On the average number of summands in partitions of n, Proc. Nat. Inst. Sci. India Part. A, 23 (1957), p. 483-498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Paul Erdős and Joseph Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8, (1941), 335-345.
John A. Ewell, Additive evaluation of the divisor function, Fibonacci Quart. 45 (2007), no. 1, 22-25. See Table 1.
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008; see p.27
I. Kessler and M. Livingston, The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.
I. Kessler and M. Livingston, The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, Graph - The asymptotic ratio
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
S. M. Luthra, On the average number of summands in partitions of n, Proc. Nat. Inst. Sci. India Part. A, 23 (1957), p. 483-498.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
C. L. Mallows & N. J. A. Sloane, Emails, Jun. 1991
Ljuben Mutafchiev, On the Largest Part Size and Its Multiplicity of a Random Integer Partition, arXiv:1712.03233 [math.PR], 2017.
Omar E. Pol, Illustration of initial terms
J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, p. 495.
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.
H. S. Wilf, A unified setting for selection algorithms (II), Annals Discrete Math., 2 (1978), 135-148.

Cf. A093694, A008284, A000005, A066633, A085410, A046746, A209423, A285220, A336902, A336903.
Main diagonal of A210485.
Column k=1 of A256193.
The version for normal multisets is A001787.
The unordered version is A001792.
The strict case is A015723.
The version for factorizations is A066637.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A336875 counts compositions with a selected part.
A339564 counts factorizations with a selected factor.
Cf. A066186, A083710, A130689, A264401, A276024, A343341.

List([0..60],n->Length(Flat(Partitions(n)))); # Muniru A Asiru, Oct 12 2018

a006128 = length . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:= add(n*x^n*mul(1/(1-x^k), k=1..n), n=1..61):
a:= n-> coeff(series(g,x,62),x,n):
seq(a(n), n=0..61);
# second Maple program:
a:= n-> add(combinat[numbpart](n-j)*numtheory[tau](j), j=1..n):
seq(a(n), n=0..61);  # Alois P. Heinz, Aug 23 2019

a[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; Table[ a[n], {n, 0, 41}]
CoefficientList[ Series[ Sum[n*x^n*Product[1/(1 - x^k), {k, n}], {n, 100}], {x, 0, 100}], x]
a[n_] := Plus @@ Max /@ IntegerPartitions@ n; Array[a, 45] (* Robert G. Wilson v, Apr 12 2011 *)
Join[{0}, ((Log[1 - x] + QPolyGamma[1, x])/(Log[x] QPochhammer[x]) + O[x]^60)[[3]]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Length /@ Table[IntegerPartitions[n] // Flatten, {n, 50}] (* Shouvik Datta, Sep 12 2021 *)

f(n)= {local(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]1,i--;s+=i*(v[i]=(n-s)\i));t+=sum(k=1,n,v[k]));t } /* Thomas Baruchel, Nov 07 2005 */

a(n) = sum(m=1, n, numdiv(m)*numbpart(n-m)) \\ Michel Marcus, Jul 13 2013

from sympy import divisor_count, npartitions
def a(n): return sum([divisor_count(m)*npartitions(n - m) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017

G.f.: Sum_{n>=1} n*x^n / Product_{k=1..n} (1-x^k).
G.f.: Sum_{k>=1} x^k/(1-x^k) / Product_{m>=1} (1-x^m).
a(n) = Sum_{k=1..n} k*A008284(n, k).
a(n) = Sum_{m=1..n} of the number of divisors of m * number of partitions of n-m.
Note that the formula for the above comment is a(n) = Sum_{m=1..n} d(m)*p(n-m) = Sum_{m=1..n} A000005(m)*A000041(n-m), if n >= 1. - Omar E. Pol, Jan 21 2013
Erdős and Lehner show that if u(n) denotes the average largest part in a partition of n, then u(n) ~ constant*sqrt(n)*log n.
a(n) = A066897(n) + A066898(n), n>0. - Reinhard Zumkeller, Mar 09 2012
a(n) = A066186(n) - A196087(n), n >= 1. - Omar E. Pol, Apr 22 2012
a(n) = A194452(n) + A024786(n+1). - Omar E. Pol, May 19 2012
a(n) = A000203(n) + A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{m=1..p(n)} A194446(m) = Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1. - Omar E. Pol, May 12 2013
a(n) = A198381(n) + A026905(n), n >= 1. - Omar E. Pol, Aug 10 2013
a(n) = O(sqrt(n)*log(n)*p(n)), where p(n) is the partition function A000041(n). - Peter Bala, Dec 23 2013
a(n) = Sum_{m=1..n} A006218(m)*A002865(n-m), n >= 1. - Omar E. Pol, Jul 14 2014
From Vaclav Kotesovec, Jun 23 2015: (Start)
Asymptotics (Luthra, 1957): a(n) = p(n) * (C*N^(1/2) + C^2/2) * (log(C*N^(1/2)) + gamma) + (1+C^2)/4 + O(N^(-1/2)*log(N)), where N = n - 1/24, C = sqrt(6)/Pi, gamma is the Euler-Mascheroni constant A001620 and p(n) is the partition function A000041(n).
The formula a(n) = p(n) * (sqrt(3*n/(2*Pi)) * (log(n) + 2*gamma - log(Pi/6)) + O(log(n)^3)) in the abstract of the article by Kessler and Livingston (cited also in the book by Sandor, p. 495) is incorrect!
Right is: a(n) = p(n) * (sqrt(3*n/2)/Pi * (log(n) + 2*gamma - log(Pi^2/6)) + O(log(n)^3))
or a(n) ~ exp(Pi*sqrt(2*n/3)) * (log(6*n/Pi^2) + 2*gamma) / (4*Pi*sqrt(2*n)).
(End)
G.f.: (log(1-x) + psi_x(1))/(log(x) * (x)inf), where psi_q(z) is the q-digamma function, and (q)_inf is the q-Pochhammer symbol (the Euler function). - _Vladimir Reshetnikov, Nov 17 2016
a(n) = Sum_{m=1..n} A341062(m)*A000070(n-m), n >= 1. - Omar E. Pol, Feb 05 2021 2014

A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n.

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 12, 5, 2, 1, 20, 8, 4, 2, 1, 35, 16, 8, 4, 2, 1, 54, 24, 13, 7, 4, 2, 1, 86, 41, 22, 13, 7, 4, 2, 1, 128, 61, 35, 20, 12, 7, 4, 2, 1, 192, 95, 54, 33, 20, 12, 7, 4, 2, 1, 275, 136, 80, 49, 31, 19, 12, 7, 4, 2, 1, 399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1, 556, 284

Offset: 1

Wouter Meeussen, Oct 09 2010

For the connection with A066897 and A066898 see A206563. - Omar E. Pol, Feb 13 2012
T(n,k) is also the total number of parts >= k in all partitions of n. - Omar E. Pol, Feb 14 2012
The first differences of row n together with 1 give the row n of triangle A066633. - Omar E. Pol, Feb 26 2012
We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1. It appears that the sum of the k-th ranks of all partitions of n is equal to zero. - Omar E. Pol, Mar 04 2012
T(n,k) is also the total number of divisors >= k of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

			From _Omar E. Pol_, Feb 13 2012: (Start)
Illustration of initial terms. First five rows of triangle as sums of columns from the partitions of the first five positive integers:
.
.                            5
.                            3+2
.                  4         4+1
.                  2+2       2+2+1
.          3       3+1       3+1+1
.     2    2+1     2+1+1     2+1+1+1
.  1  1+1  1+1+1   1+1+1+1   1+1+1+1+1
. -------------------------------------
.  1, 3,1, 6,2,1, 12,5,2,1, 20,8,4,2,1 --> This triangle
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|
.  1, 2,1, 4,1,1,  7,3,1,1, 12,4,2,1,1 --> A066633
.
For more information see A207031 and A206563.
...
Triangle begins:
    1;
    3,   1;
    6,   2,   1;
   12,   5,   2,  1;
   20,   8,   4,  2,  1;
   35,  16,   8,  4,  2,  1;
   54,  24,  13,  7,  4,  2,  1;
   86,  41,  22, 13,  7,  4,  2,  1;
  128,  61,  35, 20, 12,  7,  4,  2, 1;
  192,  95,  54, 33, 20, 12,  7,  4, 2, 1;
  275, 136,  80, 49, 31, 19, 12,  7, 4, 2, 1;
  399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1;
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Row sums are A066186. First column is A006128. Reverse of each row converges to A000070.
Columns 2-3: A096541, A207033. - Omar E. Pol, Feb 18 2012
T(2n,n) gives A216053(n+1).
Cf. A206283.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p(p(f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n) local j, l, r, t;
      l, r, t:= b(n, n), 1, 1;
      for j from n to 2 by -1 do t:= t+l[j]; r:=r, t od;
      seq([r][1+n-j], j=1..n)
    end:
seq(T(n), n=1..14); # Alois P. Heinz, Apr 05 2012

Table[Plus @@ (PadRight[ #,n]& /@ IntegerPartitions[n]),{n,16}]
(* Second program: *)
T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; Table[Table[T[n, k], {k, n, 1, -1}] // Accumulate // Reverse, {n, 1, 16}] // Flatten (* Jean-François Alcover, Oct 10 2015, after Omar E. Pol *)

T(n,k) = Sum_{j=1..n} A207031(j,k). - Omar E. Pol, May 02 2012

Better definition from Omar E. Pol, Feb 13 2012

A066897 Total number of odd parts in all partitions of n.

Original entry on oeis.org

1, 2, 5, 8, 15, 24, 39, 58, 90, 130, 190, 268, 379, 522, 722, 974, 1317, 1754, 2330, 3058, 4010, 5200, 6731, 8642, 11068, 14076, 17864, 22528, 28347, 35490, 44320, 55100, 68355, 84450, 104111, 127898, 156779, 191574, 233625, 284070, 344745, 417292, 504151

Offset: 1

Naohiro Nomoto, Jan 24 2002

Also sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n (Cf. A206563). - Omar E. Pol, Feb 12 2012
Column 1 of A206563. - Omar E. Pol, Feb 15 2012
Suppose that p=[p(1),p(2),p(3),...] is a partition of n with parts in nonincreasing order.  Let f(p) = p(1) - p(2) + p(3) - ... be the alternating sum of parts of p and let F(n) = sum of alternating sums of all partitions of n.  Conjecture: F(n) = A066897(n) for n >= 1. - Clark Kimberling, May 17 2019
From Omar E. Pol, Apr 02 2023: (Start)
Convolution of A000041 and A001227.
Convolution of A002865 and A060831.
a(n) is also the total number of odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)
a(n) is odd iff n is a term of A067567 (proof: n*p(n) = the sum of the parts in all the partitions of n == the number of odd parts in all partitions of n (mod 2). Hence the number of odd parts in all partitions of n is odd iff n*p(n) is odd, equivalently, iff both n and p(n) are odd). - Peter Bala, Jan 11 2025

			a(4) = 8 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], we have a total of 0+2+0+2+4=8 odd parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A000041, A001227, A001620, A002865, A006128, A060831, A066898, A066966, A066967, A103919, A206563, A207381, A207382, A209423, A338156.

a066897 = p 0 1 where
   p o _             0 = o
   p o k m | m < k     = 0
           | otherwise = p (o + mod k 2) k (m - k) + p o (k + 1) m
-- Reinhard Zumkeller, Mar 09 2012

a066897 = length . filter odd . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:=sum(x^(2*j-1)/(1-x^(2*j-1)),j=1..70)/product(1-x^j,j=1..70): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..44);
# Emeric Deutsch, Mar 13 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
(* Clark Kimberling, Mar 08 2012 *)
b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 26 2015, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A001227(k)=number of odd divisors of k and numbpart() is A000041. - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k=0..n} k*A103919(n,k). - Emeric Deutsch, Mar 13 2006
G.f.: Sum_{j>=1}(x^(2j-1)/(1-x^(2j-1)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Mar 13 2006
a(n) = A066898(n) + A209423(n) = A006128(n) - A066898(n). [Reinhard Zumkeller, Mar 09 2012]
a(n) = A207381(n) - A207382(n). - Omar E. Pol, Mar 11 2012
a(n) = (A006128(n) + A209423(n))/2. - Vaclav Kotesovec, May 25 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(24*n/Pi^2)) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

More terms from Vladeta Jovovic, Jan 26 2002

A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 8, 4, 1, 1, 15, 5, 3, 1, 1, 24, 11, 5, 3, 1, 1, 39, 15, 9, 4, 3, 1, 1, 58, 28, 13, 9, 4, 3, 1, 1, 90, 38, 23, 12, 8, 4, 3, 1, 1, 130, 62, 33, 21, 12, 8, 4, 3, 1, 1, 190, 85, 51, 29, 20, 11, 8, 4, 3, 1, 1, 268, 131, 73, 48, 28, 20, 11, 8, 4, 3, 1, 1

Offset: 1

Omar E. Pol, Feb 15 2012

Let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the section example. (Cf. A135010, A207031, A207032, A212010). - Omar E. Pol, May 01 2012

			Calculation for n = 6. Write the partitions of 6 and below the sums of their columns:
.
.   6
.   3 + 3
.   4 + 2
.   2 + 2 + 2
.   5 + 1
.   3 + 2 + 1
.   4 + 1 + 1
.   2 + 2 + 1 + 1
.   3 + 1 + 1 + 1
.   2 + 1 + 1 + 1 + 1
.   1 + 1 + 1 + 1 + 1 + 1
. ------------------------
.  35, 16,  8,  4,  2,  1  --> Row 6 of triangle A181187.
.   |  /|  /|  /|  /|  /|
.   | / | / | / | / | / |
.   |/  |/  |/  |/  |/  |
.  19,  8,  4,  2,  1,  1  --> Row 6 of triangle A066633.
.
More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurrences of k in all partitions of n.
On the other hand we can see that the partitions of 6 contain:
24  odd parts >= 1 (the odd parts).
11 even parts >= 2 (the even parts).
5   odd parts >= 3.
3  even parts >= 4.
2   odd parts >= 5.
1  even part  >= 6.
Then, using the values of the column sums, it appears that:
T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24
T(6,2) =      16 - 8 + 4 - 2 + 1 = 11
T(6,3) =           8 - 4 + 2 - 1 = 5
T(6,4) =               4 - 2 + 1 = 3
T(6,5) =                   2 - 1 = 1
T(6,6) =                       1 = 1
So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1.
Finally, for all partitions of 6, we can write:
The number of  odd parts      is equal to T(6,1) = 24.
The number of even parts      is equal to T(6,2) = 11.
The number of  odd parts >= 3 is equal to T(6,3) = 5.
The number of even parts >= 4 is equal to T(6,4) = 3.
The number of  odd parts >= 5 is equal to T(6,5) = 1.
The number of even parts >= 6 is equal to T(6,6) = 1.
More generally, we can write the same properties for any positive integer.
Triangle begins:
1;
2,    1;
5,    1,  1;
8,    4,  1,  1;
15,   5,  3,  1,  1;
24,  11,  5,  3,  1,  1;
39,  15,  9,  4,  3,  1,  1;
58,  28, 13,  9,  4,  3,  1,  1;
90,  38, 23, 12,  8,  4,  3,  1,  1;
130, 62, 33, 21, 12,  8,  4,  3,  1,  1;

Columns 1-2 give A066897, A066898.

Cf. A006128, A066633, A181187, A182703, A207031, A207032.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A181187(n,j)).

It appears that A066633(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

More terms from Alois P. Heinz, Feb 18 2012

A066967 Total sum of odd parts in all partitions of n.

Original entry on oeis.org

1, 2, 7, 10, 23, 36, 65, 94, 160, 230, 356, 502, 743, 1030, 1480, 2006, 2797, 3760, 5120, 6780, 9092, 11902, 15701, 20350, 26508, 34036, 43860, 55822, 71215, 89988, 113792, 142724, 179137, 223230, 278183, 344602, 426687, 525616, 647085, 792950

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206435. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 01 2023: (Start)
Convolution of A000041 and A000593.
Convolution of A002865 and A078471.
a(n) is also the sum of all odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the odd parts is (3+1)+(1+1)+(1+1+1+1) = 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000593, A066897, A066898, A113685, A206435.

Cf. A002865, A078471.

g:=sum((2*i-1)*x^(2*i-1)/(1-x^(2*i-1)),i=1..50)/product(1-x^j,j=1..50): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..47);
# Emeric Deutsch, Feb 19 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[(2*i-1)*x^(2*i-1)/(1-x^(2*i-1)), {i, 1, max}]/Product[1-x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], OddQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.
a(n) = sum(k*A113685(n,k), k=0..n). - Emeric Deutsch, Feb 19 2006
G.f.: sum((2i-1)x^(2i-1)/(1-x^(2i-1)), i=1..infinity)/product(1-x^j, j=1..infinity). - Emeric Deutsch, Feb 19 2006
a(n) = A066186(n) - A066966(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 0, 1, 7, 1, 2, 0, 1, 9, 6, 2, 2, 0, 1, 15, 4, 4, 1, 2, 0, 1, 19, 13, 4, 5, 1, 2, 0, 1, 32, 10, 10, 3, 4, 1, 2, 0, 1, 40, 24, 10, 9, 4, 4, 1, 2, 0, 1, 60, 23, 18, 8, 8, 3, 4, 1, 2, 0, 1, 78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1

Offset: 1

Omar E. Pol, Feb 17 2012

For the calculation of row n, the number of odd/even parts, etc, take the row n from the triangle A207031 and then follow the same rules of A206563.

			Triangle begins:
  1;
  1,   1;
  3,   0,  1;
  3,   3,  0,  1;
  7,   1,  2,  0, 1;
  9,   6,  2,  2, 0, 1;
  15,  4,  4,  1, 2, 0, 1;
  19, 13,  4,  5, 1, 2, 0, 1;
  32, 10, 10,  3, 4, 1, 2, 0, 1;
  40, 24, 10,  9, 4, 4, 1, 2, 0, 1;
  60, 23, 18,  8, 8, 3, 4, 1, 2, 0, 1;
  78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1;

Cf. A006128, A066897, A066898, A135010, A138121, A138135, A138137, A141285, A181187, A182703, A206563, A207031.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A207031(n,j)).

It appears that A182703(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.

Original entry on oeis.org

1, 1, 4, 4, 10, 13, 24, 30, 52, 68, 105, 137, 202, 264, 376, 485, 669, 864, 1162, 1486, 1968, 2501, 3256, 4110, 5285, 6630, 8434, 10511, 13241, 16417, 20505, 25273, 31344, 38438, 47346, 57782, 70746, 85947, 104663, 126594, 153386, 184793, 222865, 267452

Offset: 1

Clark Kimberling, Mar 08 2012

nonn

a(n) = number of parts of odd multiplicity (each counted only once) in all partitions of n. Example: a(5) = 10 because we have [5'],[4',1'],[3',2'], [3',1,1],[2,2,1'],[2',1',1,1], and [1',1,1,1,1] (the 10 counted parts are marked). - Emeric Deutsch, Feb 08 2016

			The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1], a total of 15 odd parts and 5 even parts, so that a(5)=10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A066897, A066898, A000041, A240009, A264398.

b:= proc(n, i) option remember; local m, f, g;
      m:= irem(i, 2);
      if n=0 then [1, 0, 0]
    elif i<1 then [0, 0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$3], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+m*g[1], f[3]+g[3]+(1-m)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n, n)[3]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012
g := add(x^j/(1+x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 08 2016

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
b[n_, i_] := b[n, i] = Module[{m, f, g}, m = Mod[i, 2]; If[n==0, {1, 0, 0}, If[i<1, {0, 0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + m*g[[1]], f[[3]] + g[[3]] + (1-m)* g[[1]]}]]]; a[n_] := b[n, n][[2]] - b[n, n][[3]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

a(n) = A066897(n) - A066898(n) = A206563(n,1) - A206563(n,2). - Omar E. Pol, Mar 08 2012
G.f.: Sum_{j>0} x^j/(1+x^j)/Product_{k>0}(1 - x^k). - Emeric Deutsch, Feb 08 2016
a(n) = Sum_{i=1..n} (-1)^(i + 1)*A181187(n, i). - John M. Campbell, Mar 18 2018
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * Pi * sqrt(n)). - Vaclav Kotesovec, May 25 2018
For n > 0, a(n) = A305121(n) + A305123(n). - Vaclav Kotesovec, May 26 2018
a(n) = Sum_{k=-floor(n/2)+(n mod 2)..n} k * A240009(n,k). - Alois P. Heinz, Oct 23 2018
a(n) = Sum_{k>0} k * A264398(n,k). - Alois P. Heinz, Aug 05 2020

A067588 Total number of parts in all partitions of n into odd parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 14, 19, 26, 36, 48, 62, 82, 104, 132, 169, 210, 260, 324, 396, 484, 592, 714, 860, 1036, 1238, 1474, 1756, 2078, 2452, 2894, 3396, 3976, 4654, 5422, 6309, 7332, 8490, 9816, 11338, 13060, 15018, 17254, 19774, 22630, 25878, 29524, 33642

Offset: 0

Naohiro Nomoto, Jan 31 2002

Starting with "1" = triangle A097304 * [1, 2, 3, ...]. - Gary W. Adamson, Apr 09 2010

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 17-34.

Cf. A067589, A006128, A066897, A066898.

Cf. A097304. - Gary W. Adamson, Apr 09 2010

G.f.: G(x)*H(x) where G(x) = Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) is g.f. for the number of odd divisors of n (cf. A001227) and H(x) = Product_{k>=1} (1+x^k) is g.f. for the number of partitions of n into odd parts (cf. A000009). Convolution of A001227 and A000009: Sum_{k=0..n} A001227(k)*A000009(n-k). - Vladeta Jovovic, Feb 04 2002
G.f.: Sum_{n>0} n*x^n/Product_{k=1..n} (1-x^(2*k)). - Vladeta Jovovic, Dec 15 2003
a(n) ~ 3^(1/4) * (2*gamma + log(48*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (8*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

Corrected by James Sellers, May 31 2007

A066966 Total sum of even parts in all partitions of n.

Original entry on oeis.org

0, 2, 2, 10, 12, 30, 40, 82, 110, 190, 260, 422, 570, 860, 1160, 1690, 2252, 3170, 4190, 5760, 7540, 10142, 13164, 17450, 22442, 29300, 37410, 48282, 61170, 78132, 98310, 124444, 155582, 195310, 242722, 302570, 373882, 462954, 569130, 700570, 856970

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206436. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 02 2023: (Start)
Convolution of A000041 and A146076.
Convolution of A002865 and A271342.
a(n) is also the sum of all even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the even parts is 4+2+2+2 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000203, A002865, A066186, A066897, A066898, A066967, A113686, A146076, A206436, A271342, A338156.

g:=sum(2*j*x^(2*j)/(1-x^(2*j)),j=1..55)/product(1-x^j,j=1..55): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..41);
# Emeric Deutsch, Feb 20 2006
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, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[2*j*x^(2*j)/(1 - x^(2*j)), {j, 1, max}]/Product[1 - x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max - 1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ Joerg Arndt, Jan 24 2014

a(n) = 2*Sum_{k=1..floor(n/2)} sigma(k)*numbpart(n-2*k).
a(n) = Sum_{k=0..n} k*A113686(n,k). - Emeric Deutsch, Feb 20 2006
G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Feb 20 2006
a(n) = A066186(n) - A066967(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

More terms from Emeric Deutsch, Feb 20 2006

A116482 Triangle read by rows: T(n,k) is the number of partitions of n having k even parts (n>=0, 0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 6, 3, 1, 6, 8, 5, 2, 1, 8, 11, 7, 3, 1, 10, 14, 10, 5, 2, 1, 12, 19, 14, 7, 3, 1, 15, 24, 19, 11, 5, 2, 1, 18, 31, 26, 15, 7, 3, 1, 22, 39, 34, 21, 11, 5, 2, 1, 27, 49, 45, 29, 15, 7, 3, 1, 32, 61, 58, 39, 22, 11, 5, 2, 1, 38, 76, 75, 52, 30

Offset: 0

Emeric Deutsch, Feb 17 2006

Row n has 1 + floor(n/2) terms. Row sums are the partition numbers (A000041).
Column 0 yields A000009. Column 1 yields A038348. Column 2 yields A096778.
Sum_{k=0..floor(n/2)}k*T(n,k) = A066898(n).
From Gregory L. Simay, Nov 02 2015: (Start)
If n<=2k+1, T(n+2k,k) = A000041(n), the number of partitions of n.
T(n+2k,k) = the convolution of A000009(n-2j),which are the strict partitions of (n-2j), and p(j+k,k), which are the number of partitions of j+k having exactly k parts.
T(n+2k,k) = e(n,k) where e(n,0)= A000009(n) and e(n,k) = e(n,k-1) + e(n-2k,k-1) + e(n-4k,k-1) + ... .(End)

			T(7,2) = 3 because we have [4,2,1], [3,2,2] and [2,2,1,1,1].
Triangle starts:
   1;
   1;
   1,  1;
   2,  1;
   2,  2,  1;
   3,  3,  1;
   4,  4,  2,  1;
   5,  6,  3,  1;
   6,  8,  5,  2,  1;
   8, 11,  7,  3,  1;
  10, 14, 10,  5,  2, 1;
  12, 19, 14,  7,  3, 1;
  15, 24, 19, 11,  5, 2, 1;
  18, 31, 26, 15,  7, 3, 1;
  22, 39, 34, 21, 11, 5, 2, 1;
  27, 49, 45, 29, 15, 7, 3, 1;
Added entries for n=8 through n=15. - _Gregory L. Simay_, Nov 03 2015
From _Gregory L. Simay_, Nov 03 2015: (Start)
T(15,4) = T(7+2*4,4) = p(7) = 15, since 7 < 2*4 + 1.
T(15,3) = T(13,2) + T(9,3) = 26 + 3 = 29.
T(10,1) = T(8+2*1,1) = T(8,0) + T(6,0) + T(4,0) + T(2,0) + T(0,0) = 6 + 4 + 2 + 1 + 1 = 14.
T(15,3) = T(9+2*3) = e(9,3) = e(9,2) + e(3,2) = (e(9,1) + e(5,1) + e(1,1)) + e(3,1) = q(9) + q(7) + q(5) + q(3) + q(1) + q(5) + q(3) + q(1) + q(1) + q(3) + q(1) = q(9) + q(7) + 2*q(5) + 3*q(3) + 4*q(1) = 8 + 5 + 2*3 + 3*2 + 4*1 = 29 = the convolution sum of q(9-2j) with p(3+j,3).
(End)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A038348, A066898, A103919.

g:=1/product((1-x^(2*j-1))*(1-t*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1
      then 0 else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(irem(i, 2)=0, 0$j, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq (T(n), n=0..30);  # Alois P. Heinz, Jan 07 2013

nn=8;CoefficientList[Series[Product[1/(1-x^(2i-1))/(1-y x^(2i)),{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Jan 07 2013 *)

G.f.: G(t,x) = 1/Product_{j>=1}((1-x^(2j-1))(1-tx^(2j))).
From Gregory L. Simay, Nov 03 2015: (Start)
G.f.: T(n+2k,k) = g.f.: e(n,k) = Product_{j>=1}(1-x^2*(k+j))*p(x), where p(x) is the g.f. of the partitions of x. If n<=2k+1, then the g.f. reduces to p(x).
T(n+2k,k) = T(n+2k-2,k-1) + T(n,k).
(End)

cp's OEIS Frontend

A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n.

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A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n.

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A066897 Total number of odd parts in all partitions of n.

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A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

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A066967 Total sum of odd parts in all partitions of n.

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A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.

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