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A218700
Number of partitions of n in which any two distinct parts differ by at least 5.
2
1, 1, 2, 2, 3, 2, 4, 3, 6, 7, 9, 10, 15, 15, 19, 23, 26, 28, 36, 37, 48, 52, 62, 67, 85, 93, 110, 122, 144, 157, 194, 205, 241, 265, 304, 338, 391, 422, 483, 533, 607, 661, 760, 822, 933, 1032, 1151, 1260, 1432, 1554, 1751, 1920, 2137, 2333, 2621, 2848, 3176
OFFSET
0,3
COMMENTS
Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 5 times.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(5*i)/(1-x^i)).
log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-5*x)) dx = -0.9908078441778424729564846063206238729218368028... - Vaclav Kotesovec, Jan 28 2022
EXAMPLE
a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
a(7) = 3: [1,1,1,1,1,1,1], [1,6], [7].
a(8) = 6: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,6], [1,7], [8].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-5), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..70);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i j, i - k, k], {j, 1, n/i}]]];
a[n_] := b[n, n, 5];
a /@ Range[0, 70] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A218698.
Cf. A160975.
Sequence in context: A307995 A061889 A240089 * A325331 A333108 A266935
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 04 2012
STATUS
approved