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
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
Programs
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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
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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 *)
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
Extensions
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
Comments