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A116932 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times. 13
1, 2, 2, 3, 3, 6, 6, 9, 12, 14, 16, 24, 25, 32, 40, 49, 56, 73, 81, 102, 120, 142, 162, 202, 227, 270, 316, 367, 419, 506, 565, 663, 767, 879, 998, 1179, 1317, 1517, 1739, 1979, 2232, 2588, 2883, 3295, 3742, 4220, 4737, 5426, 6037, 6828, 7701, 8642, 9651, 10939 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also, partitions of n in which any two distinct parts differ by at least 3. Example: a(5) = 3 because we have [5], [4,1] and [1,1,1,1,1].
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
FORMULA
G.f.: sum(x^k*product(1+x^(3j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.
log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869... - Vaclav Kotesovec, Jan 28 2022
EXAMPLE
a(5) = 3 because we have [5], [2,1,1,1] and [1,1,1,1,1].
MAPLE
g:=sum(x^k*product(1+x^(3*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 62): seq(coeff(gser, x^n), n=1..58);
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-3), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=1..70); # Alois P. Heinz, Nov 04 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-3], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)
CROSSREFS
Column k=3 of A218698. - Alois P. Heinz, Nov 04 2012
Sequence in context: A133392 A101199 A032155 * A240579 A292225 A238786
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 27 2006
STATUS
approved

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Last modified June 23 05:14 EDT 2024. Contains 373629 sequences. (Running on oeis4.)