A079499 Total number of parts in all partitions of n into distinct odd parts.

0, 1, 0, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 7, 6, 10, 12, 13, 12, 16, 18, 22, 22, 25, 32, 36, 36, 42, 50, 53, 58, 64, 76, 83, 88, 99, 116, 123, 132, 147, 168, 181, 194, 215, 240, 262, 280, 306, 346, 375, 396, 437, 482, 521, 558, 610, 670, 724, 772, 840, 922, 993, 1056, 1151, 1256, 1348

Offset: 0

Author

Arnold Knopfmacher, Jan 21 2003

Keywords

• nonn

Comments

Also sum of the sizes of the Durfee squares of all self-conjugate partitions of n. Example: a(13)=7 because there are three self-conjugate partitions of 13, namely [7,1,1,1,1,1,1], [5,3,3,1,1] and [4,4,3,2], having Durfee squares of sizes 1,3 and 3, respectively. a(n) = Sum_{k=1..floor(sqrt(n))} k*A116422(n,k). - Emeric Deutsch, Feb 14 2006

Examples

			a(13)=7 because the partitions of 13 into distinct odd parts are [13], [9,3,1] and [7,5,1] and we have 1+3+3=7 parts.
		

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
• G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
• Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Crossrefs

Cf. A015723, A000700, A067619, A006128.

Cf. A032021.

Cf. A116422.

Programs

Maple

g:=sum(k*x^(k^2)/product(1-x^(2*i),i =1..k),k=1..20):gser:=series(g,x=0,52): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Feb 14 2006

Mathematica

max = 100; s = Sum[ k*x^(k^2) / Product[1-x^(2*j), {j, 1, k}], {k, 1, Sqrt[max] // Ceiling}]; CoefficientList[ Series[s, {x, 0, max}], x] (* Jean-François Alcover, Feb 19 2015, after Vladeta Jovovic *)

PARI

N=66;  S=2+sqrtint(N); x='x+O('x^N);
gf=sum(n=0, S, n*x^(n^2)/prod(k=1,n, 1-x^(2*k)) );
concat( [0], Vec(gf) )
\\ Joerg Arndt, Feb 18 2014

Formula

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

G.f.: Sum_{k>=1} k*x^(k^2)/Product_{j=1..k} (1 - x^(2*j)). - Vladeta Jovovic, Aug 06 2004

a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/6)) / (Pi * 2^(5/4) * n^(1/4)). - Vaclav Kotesovec, May 20 2018