A092316 Sum of largest parts of all partitions of n into odd distinct parts.

1, 0, 3, 3, 5, 5, 7, 12, 14, 16, 18, 27, 29, 33, 42, 55, 59, 65, 78, 95, 110, 118, 137, 167, 188, 200, 236, 274, 303, 330, 376, 435, 485, 522, 591, 677, 741, 803, 903, 1022, 1115, 1210, 1345, 1505, 1650, 1784, 1964, 2201, 2393, 2578, 2843, 3143, 3409, 3685, 4034

Offset: 1

Author

Vladeta Jovovic, Feb 15 2004

Keywords

• easy
• nonn

Examples

			a(13) = 29 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of largest terms 13+9+7 = 29.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
• Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Crossrefs

Cf. A000700, A005895, A067619, A092319, A116857.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or i^2 `if`(t>n, 0, b(n-t, i-1)))(2*i-1) ))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, (j-1)/2), 0), j=1..n):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 19 2022

Mathematica

nmax = 50; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j - 1), {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

Formula

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

a(n) = 2 * A067619(n) - A000700(n). - Seiichi Manyama, Jan 19 2022

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004