A209423 -id:A209423 - 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

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

A240009 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 1, 1, 2, 2, 2, 4, 3, 2, 2, 1, 1, 0, 1, 1, 2, 4, 5, 3, 4, 4, 2, 2, 1, 1, 0, 1, 1, 1, 2, 3, 3, 5, 7, 5, 4, 4, 2, 2, 1, 1, 0, 1, 1, 2, 4, 7, 7, 6, 8, 6, 4, 4, 2, 2, 1, 1, 0, 1

Offset: 0

Alois P. Heinz, Mar 30 2014

T(n,k) = T(n+k,-k).
Sum_{k=-floor(n/2)+(n mod 2)..-1} T(n,k) = A108949(n).
Sum_{k=-floor(n/2)+(n mod 2)..0} T(n,k) = A171966(n).
Sum_{k=1..n} T(n,k) = A108950(n).
Sum_{k=0..n} T(n,k) = A130780(n).
Sum_{k=-1..1} T(n,k) = A239835(n).
Sum_{k<>0} T(n,k) = A171967(n).
T(n,-1) + T(n,1) = A239833(n).
Sum_{k=-floor(n/2)+(n mod 2)..n} k * T(n,k) = A209423(n).
Sum_{k=-floor(n/2)+(n mod 2)..n} (-1)^k*T(n,k) = A081362(n) = (-1)^n*A000700(n).

			T(5,-1) = 1: [2,2,1].
T(5,0) = 2: [4,1], [3,2].
T(5,1) = 1: [5].
T(5,2) = 1: [2,1,1,1].
T(5,3) = 1: [3,1,1].
T(5,5) = 1: [1,1,1,1,1].
Triangle T(n,k) begins:
: n\k : -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 ...
+-----+----------------------------------------------------
:  0  :                 1;
:  1  :                    1;
:  2  :              1, 0, 0, 1;
:  3  :                 1, 1, 0, 1;
:  4  :           1, 1, 0, 1, 1, 0, 1;
:  5  :              1, 2, 1, 1, 1, 0, 1;
:  6  :        1, 1, 1, 1, 2, 2, 1, 1, 0, 1;
:  7  :           1, 2, 3, 2, 2, 2, 1, 1, 0, 1;
:  8  :     1, 1, 2, 2, 2, 4, 3, 2, 2, 1, 1, 0, 1;
:  9  :        1, 2, 4, 5, 3, 4, 4, 2, 2, 1, 1, 0, 1;
: 10  :  1, 1, 2, 3, 3, 5, 7, 5, 4, 4, 2, 2, 1, 1, 0, 1;

Alois P. Heinz, Rows n = 0..120, flattened

Columns k=(-1)-10 give: A239832, A045931, A240010, A240011, A240012, A240013, A240014, A240015, A240016, A240017, A240018, A240019.
Row sums give A000041.
T(2n,n) gives A002865.
T(4n,2n) gives A182746.
T(4n+2,2n+1) gives A182747.
Row lengths give A016777(floor(n/2)).
Cf. A240021 (the same for partitions into distinct parts), A242618 (the same for parts counted without multiplicity).
Cf. A000700, A081362, A209423.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i)*x^(2*irem(i, 2)-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]*x^(2*Mod[i, 2]-1)]]]; T[n_] := (degree = Exponent[b[n, n], x]; ldegree = -Exponent[b[n, n] /. x -> 1/x, x]; Table[Coefficient[b[n, n], x, i], {i, ldegree, degree}]); Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)

N=20; q='q+O('q^N);
e(n) = if(n%2!=0, u, 1/u);
gf = 1 / prod(n=1,N, 1 - e(n)*q^n );
V = Vec( gf );
{ for (j=1, #V,  \\ print triangle, including leading zeros
    for (i=0, N-j, print1("   "));  \\ padding
    for (i=-j+1, j-1, print1(polcoeff(V[j], i, u),", "));
    print();
); }
/* Joerg Arndt, Mar 31 2014 */

G.f.: 1 / prod(n>=1, 1 - e(n)*q^n ) = 1 + sum(n>=1, e(n)*q^n / prod(k=1..n, 1-e(k)*q^k) ) where e(n) = u if n odd, otherwise 1/u; see Pari program. [Joerg Arndt, Mar 31 2014]

A066898 Total number of even parts in all partitions of n.

Original entry on oeis.org

0, 1, 1, 4, 5, 11, 15, 28, 38, 62, 85, 131, 177, 258, 346, 489, 648, 890, 1168, 1572, 2042, 2699, 3475, 4532, 5783, 7446, 9430, 12017, 15106, 19073, 23815, 29827, 37011, 46012, 56765, 70116, 86033, 105627, 128962, 157476, 191359, 232499, 281286, 340180, 409871

Offset: 1

Naohiro Nomoto, Jan 24 2002

Also sum of all even-indexed parts minus the sum of all odd-indexed parts, except the largest parts, of all partitions of n (cf. A206563). - Omar E. Pol, Feb 14 2012
From Omar E. Pol, Apr 06 2023: (Start)
Convolution of A000041 and A183063.
Convolution of A002865 and A362059.
a(n) is also the total number of 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(5) = 5 because in all the partitions of 5, namely [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], we have a total of 0+1+1+0+2+1+0=5 even parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

Cf. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.

Column 2 of A206563.

a066898 = p 0 1 where
   p e _             0 = e
   p e k m | m < k     = 0
           | otherwise = p (e + 1 - mod k 2) k (m - k) + p e (k + 1) m
-- Reinhard Zumkeller, Mar 09 2012

a066898 = length . filter even . 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-x^(2*j)),j=1..60)/product((1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50); # Emeric Deutsch, Feb 17 2006
A066898 := proc(n)
    add(numtheory[tau](k)*combinat[numbpart](n-2*k),k=1..n/2) ;
end proc: # R. J. Mathar, Jun 18 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 *)
(* Clark Kimberling, Mar 08 2012 *)
a[n_] := Sum[DivisorSigma[0, k] PartitionsP[n - 2k], {k, 1, n/2}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 31 2016, after Vladeta Jovovic *)

a(n) = Sum_{k=1..floor(n/2)} tau(k)*numbpart(n-2*k). - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k=0..floor(n/2)} k*A116482(n,k). - Emeric Deutsch, Feb 17 2006
G.f.: (Sum_{j>=1} x^(2*j)/(1-x^(2*j)))/(Product_{j>=1} (1-x^j)). - Emeric Deutsch, Feb 17 2006
a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). - Reinhard Zumkeller, Mar 09 2012
a(n) = (A006128(n) - A209423(n))/2. - Vaclav Kotesovec, May 25 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(3*n/(2*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

A305102 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 4, 10, 23, 46, 88, 158, 274, 459, 748, 1190, 1858, 2846, 4292, 6384, 9373, 13602, 19536, 27782, 39158, 54740, 75928, 104562, 143036, 194423, 262704, 352988, 471778, 627382, 830352, 1093994, 1435132, 1874920, 2439832, 3163020, 4085825, 5259602, 6748136

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A006128 and A000009.
Convolution of A305082 and A000041.
Convolution of A000005 and A015128.
a(n) is the number of non-overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006128, A015723, A209423, A305082, A305101.

Cf. A335651 and A335666.

nmax = 40; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, 1*q^k/(1-q^k)) ) \\ Joerg Arndt, Jun 18 2020

a(n) ~ exp(Pi*sqrt(n)) * (2*gamma + log(4*n/Pi^2)) / (8*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 39, 54, 71, 94, 124, 159, 201, 258, 322, 401, 499, 613, 750, 918, 1110, 1340, 1617, 1935, 2308, 2752, 3261, 3854, 4554, 5350, 6273, 7348, 8572, 9983, 11612, 13460, 15578, 18007, 20761, 23894, 27473, 31511, 36090, 41296, 47152, 53767

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A000005 and A000009.

Apart from initial zero this is the convolution of A341062 and A036469. - Omar E. Pol, Feb 16 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A000009, A006128, A015723, A209423.

Cf. A001227, A048272, A067588, A090867, A036469, A341062.

nmax = 50; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[((Log[1-x] + QPolyGamma[0, 1, x]) * QPochhammer[-1, x]) / (2*Log[x]), {x, 0, nmax}], x]

a(n) ~ 3^(1/4)*(2*gamma + log(12*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620.

A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 1, 0, 3, 2, 7, 6, 15, 16, 32, 36, 62, 74, 117, 142, 214, 264, 377, 468, 648, 806, 1090, 1354, 1791, 2224, 2894, 3580, 4598, 5670, 7193, 8838, 11102, 13588, 16925, 20632, 25501, 30972, 38021, 46000, 56135, 67668, 82119, 98642, 119115, 142592, 171412, 204520

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Conjecture: a(n) is odd iff n is a term of A067567. - Peter Bala, Jan 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A001227, A006128, A067567, A209423, A066898, A305121, A066897.

Cf. A116676, A305124.

nmax = 60; CoefficientList[Series[Sum[x^(2*k-1)/(1+x^(2*k-1)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305121(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 7, 9, 14, 20, 32, 43, 63, 85, 122, 162, 221, 292, 396, 514, 680, 878, 1147, 1465, 1886, 2391, 3050, 3836, 4841, 6048, 7579, 9403, 11685, 14419, 17806, 21845, 26810, 32725, 39947, 48528, 58926, 71267, 86151, 103750, 124860, 149791, 179551

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006128, A209423, A066898, A066897, A305123.

Cf. A116680, A305122.

nmax = 60; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305123(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A264398 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with odd multiplicities.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 1, 0, 4, 3, 3, 4, 3, 1, 0, 7, 7, 1, 5, 7, 7, 3, 0, 12, 14, 4, 7, 12, 14, 8, 1, 0, 19, 26, 10, 1, 11, 19, 26, 18, 3, 0, 30, 45, 22, 4, 15, 30, 45, 36, 9, 0, 45, 75, 44, 11, 1, 22, 45, 75, 67, 21, 1, 0, 67, 120, 81, 26, 3, 30, 67, 120, 119, 45, 4

Offset: 0

Emeric Deutsch, Nov 21 2015

T(n,0) = A035363(n).

Row sums give A000041. - Omar E. Pol, Nov 21 2015

			T(6,1) = 4 because we have [6*], [4*,1,1],[2*,2,2], and [2*,1,1,1,1] (parts with odd multiplicities are marked).
Triangle starts:
  1;
  0, 1;
  1, 1;
  0, 2, 1;
  2, 2, 1;
  0, 4, 3;
  3, 4, 3, 1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30

Cf. A000041, A035363, A209423, A264399, A264400.

g := product((1+t*x^j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(`if`(j::odd, x, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 25 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[OddQ[j], x, 1]* b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2016, after Alois P. Heinz *)

T(n) = { Vec(prod(k=1, n, (1 + y*x^k)/(1 - x^(2*k)) + O(x*x^n))) }
{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

G.f.: G(t,x) = Product_{j>=1} (1 + tx^j)/(1 - x^(2j)).
Sum_{k>0} k * T(n,k) = A209423(n). - Alois P. Heinz, Aug 05 2020
G.f.: A(x,y)*B(x^2) where A(x),B(x) are the o.g.f.'s for A008289 and A000041. (See Flajolet, Sedgewick link.) - Geoffrey Critzer, Aug 07 2022

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

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A240009 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

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A066898 Total number of even parts in all partitions of n.

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A305102 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

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A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).