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A320691
Number of partitions of n with up to four distinct kinds of 1.
2
1, 4, 7, 9, 13, 20, 29, 41, 57, 78, 106, 142, 188, 248, 324, 419, 540, 692, 880, 1114, 1404, 1761, 2200, 2736, 3389, 4186, 5152, 6317, 7724, 9418, 11449, 13882, 16789, 20253, 24376, 29271, 35071, 41936, 50041, 59591, 70835, 84049, 99547, 117703, 138941, 163746
OFFSET
0,2
LINKS
FORMULA
a(n) ~ Pi * 2^(3/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018
G.f.: (1 + x)^4 * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i=1,
binomial(4, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[4, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
a[n_] := b[n, n];
a /@ Range[0, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
Column k=4 of A292622.
Sequence in context: A310962 A310963 A082869 * A139444 A353020 A166569
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 19 2018
STATUS
approved