%I #18 Jan 27 2019 03:00:50

%S 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,

%T 62,62,71,66,84,91,103,96,119,127,139,137,167,178,191,192,223,241,266,

%U 264,302,331,351,360,411,439,469,485,542,587,628,646,714,773,819,854,945

%N Sum of smallest parts of all partitions of n into odd distinct parts.

%C a(n) = Sum_{k>=0} A116860(n,k). - _Emeric Deutsch_, Feb 27 2006

%H Alois P. Heinz, <a href="/A092319/b092319.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/6)) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, May 20 2018

%e 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.

%p 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

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p `if`(i>n, 0, b(n, i+2)+b(n-i, i+2)))

%p end:

%p a:= n-> add(`if`(j::odd, j*b(n-j, j+2), 0), j=1..n):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Feb 03 2016

%t 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 *)

%Y Cf. A092316.

%Y Cf. A116860.

%K easy,nonn

%O 1,3

%A _Vladeta Jovovic_, Feb 15 2004

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