OFFSET
1,3
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
FORMULA
a(n) = Sum_{k=0..n-1} k*A114087(n,k).
G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..oo)}]_{t=1}.
a(n) ~ (1/(8*sqrt(3)) - sqrt(3) * (log(2))^2 / (4*Pi^2)) * exp(Pi*sqrt(2*n/3)). - Vaclav Kotesovec, Jan 03 2019
EXAMPLE
a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having total size 0+1+0+2+3=6.
MAPLE
g:=sum(z^(k^2)/product((1-z^j)*(1-(t*z)^j), j=1..k), k=1..10): dgdt1:=simplify(subs(t=1, diff(g, t))): dgdt1ser:=series(dgdt1, z=0, 55): seq(coeff(dgdt1ser, z, n), n=1..48);
# second Maple program:
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
end:
a:= n-> add(k*add(b(k, d) *b(n-d^2-k, d),
d=0..floor(sqrt(n))), k=0..n-1):
seq(a(n), n=1..40); # Alois P. Heinz, Apr 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[k*Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}], {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 12 2006
STATUS
approved