OFFSET
0,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition (number theory)
FORMULA
From Vaclav Kotesovec, May 27 2020: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161... if mod(n,4) = 0
c = 96146521894.9433... if mod(n,4) = 1
c = 96146522937.2138... if mod(n,4) = 2
c = 96146521894.8218... if mod(n,4) = 3
(End)
EXAMPLE
a(3) = 1:
3 -> 21 -> 111
a(4) = 6:
4 -> 31 -> 211
4 -> 31 -> 1111
4 -> 22 -> 112
4 -> 22 -> 211
4 -> 22 -> 1111
4 -> 211-> 1111
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..37);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[b[n, n, i] (-1)^(2 - i) Binomial[2, i], {i, 0, 2}];
a /@ Range[0, 37] (* Jean-François Alcover, May 01 2020, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 24 2019
STATUS
approved