A116932 - OEIS

%I #21 Jan 28 2022 15:43:42

%S 1,2,2,3,3,6,6,9,12,14,16,24,25,32,40,49,56,73,81,102,120,142,162,202,

%T 227,270,316,367,419,506,565,663,767,879,998,1179,1317,1517,1739,1979,

%U 2232,2588,2883,3295,3742,4220,4737,5426,6037,6828,7701,8642,9651,10939

%N Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times.

%C 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].

%H Vaclav Kotesovec, <a href="/A116932/b116932.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%F 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.

%F 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

%e a(5) = 3 because we have [5], [2,1,1,1] and [1,1,1,1,1].

%p 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);

%p # second Maple program

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p        b(n, i-1) +add(b(n-i*j, i-3), j=1..n/i)))

%p     end:

%p a:= n-> b(n, n):

%p seq(a(n), n=1..70);  # _Alois P. Heinz_, Nov 04 2012

%t 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_ *)

%Y Cf. A116931, A100405.

%Y Column k=3 of A218698. - _Alois P. Heinz_, Nov 04 2012

%K nonn

%O 1,2

%A _Emeric Deutsch_, Feb 27 2006