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

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

0,3

Comments

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

Links

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.

Formula

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

Crossrefs

Cf. A067589, A006128, A066897, A066898.

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

Sequence in context: A113753 A024457 A117842 * A003402 A328863 A218004

Adjacent sequences: A067585 A067586 A067587 * A067589 A067590 A067591

Keyword

easy,nonn

Author

Naohiro Nomoto, Jan 31 2002

Extensions

Corrected by James A. Sellers, May 31 2007

Status

approved