0,5

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

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).

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.

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

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.

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

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

Cf. A015723, A000700, A067619, A006128.

Cf. A032021.

Cf. A116422.

Sequence in context: A380480 A099875 A383056 * A166235 A143591 A380282

Adjacent sequences: A079496 A079497 A079498 * A079500 A079501 A079502

Arnold Knopfmacher, Jan 21 2003

approved