OFFSET
0,2
COMMENTS
Using a different (forward) definition of the difference operator, this sequence has also been given as 0, 1, 6, 23, 64, 129, 222, ... A119712.
LINKS
Jean-François Alcover, Table of n, a(n) for n = 0..60
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
Charles Knessl, Asymptotic Behavior of High-Order Differences of the Partition Function, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), 237-254.
Eric Weisstein's World of Mathematics, Backward Difference
FORMULA
An asymptotic formula is a(n) ~ 6/Pi^2 * n^2 (log n)^2.
MAPLE
A41:= n-> `if` (n<0, 0, combinat[numbpart](n)):
DB:= proc(p)
proc(n) option remember;
p(n) -p(n-1)
end
end:
a:= proc(n) option remember;
local f, k;
if n=0 then 1
else f:= (DB@@n)(A41);
for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k
fi
end:
seq(a(n), n=0..20);
MATHEMATICA
a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j + 1] > 0, Return[j + n]]]];
a[0] = 1; a[1] = 2;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 16 2010
STATUS
approved