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

1,3

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.

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

Example

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.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or i^2<n,
       0, b(n, i-1)+ (t-> `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 *)

Crossrefs

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

Sequence in context: A087715 A237714 A245145 * A327710 A339101 A142456

Adjacent sequences: A092313 A092314 A092315 * A092317 A092318 A092319

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 15 2004

Extensions

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

Status

approved