

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
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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



FORMULA

G.f.: sum(x^k*product(1+x^(3j)/(1x^j), j=1..k1)/(1x^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)/(1x^j), j=1..k1)/(1x^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)/(1x^j), j=1..k1)/(1x^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, i1) +add(b(ni*j, i3), j=1..n/i)))
end:
a:= n> b(n, n):


MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i1] + Sum[b[ni*j, i3], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* JeanFrançois Alcover, May 26 2015, after Alois P. Heinz *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



