A092319 Sum of smallest parts of all partitions of n into odd distinct parts.

1, 0, 3, 1, 5, 1, 7, 4, 10, 4, 12, 9, 15, 9, 20, 17, 23, 17, 28, 27, 36, 28, 41, 43, 50, 44, 62, 62, 71, 66, 84, 91, 103, 96, 119, 127, 139, 137, 167, 178, 191, 192, 223, 241, 266, 264, 302, 331, 351, 360, 411, 439, 469, 485, 542, 587, 628, 646, 714, 773, 819, 854, 945

Offset

1,3

Comments

a(n) = Sum_{k>=0} A116860(n,k). - Emeric Deutsch, Feb 27 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum((2*n-1)*x^(2*n-1)*Product(1+x^(2*k+1), k = n .. infinity), n = 1 .. infinity).

a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/6)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

Example

a(13)=15 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of the smallest terms 13+1+1=15.

Maple

f:=sum((2*n-1)*x^(2*n-1)*product(1+x^(2*k+1), k=n..40), n=1..40): fser:=simplify(series(f, x=0, 66)): seq(coeff(fser, x^n), n=1..63); # Emeric Deutsch, Feb 27 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+2)+b(n-i, i+2)))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, j+2), 0), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

Mathematica

nmax = 60; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j + 1), {j, k, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

Crossrefs

Cf. A092316.

Cf. A116860.

Sequence in context: A136180 A095112 A160596 * A254938 A244149 A378642

Adjacent sequences: A092316 A092317 A092318 * A092320 A092321 A092322

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 15 2004

Extensions

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

Status

approved