OFFSET
1,2
COMMENTS
Also, partitions of n in which any two distinct parts differ by at least 2. Example: a(5) = 4 because we have [5], [4,1], [3,1,1] and [1,1,1,1,1].
REFERENCES
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 52, Article 298.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..20000 (terms 1..7500 from Alois P. Heinz)
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.
FORMULA
G.f.: sum(x^k*product(1+x^(2j)/(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)) ~ 2*Pi*sqrt(n)/3. - Vaclav Kotesovec, Jan 28 2022
EXAMPLE
a(5) = 4 because we have [5], [3,1,1], [2,1,1,1] and [1,1,1,1,1].
q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 13*q^8 + 15*q^9 + ...
There are a(9) = 15 partitions of 9 where distinct parts differ by at least 2:
01: [ 1 1 1 1 1 1 1 1 1 ]
02: [ 3 1 1 1 1 1 1 ]
03: [ 3 3 1 1 1 ]
04: [ 3 3 3 ]
05: [ 4 1 1 1 1 1 ]
06: [ 4 4 1 ]
07: [ 5 1 1 1 1 ]
08: [ 5 2 2 ]
09: [ 5 3 1 ]
10: [ 6 1 1 1 ]
11: [ 6 3 ]
12: [ 7 1 1 ]
13: [ 7 2 ]
14: [ 8 1 ]
15: [ 9 ]
- Joerg Arndt, Jun 09 2013
MAPLE
g:=sum(x^k*product(1+x^(2*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 60): seq(coeff(gser, x^n), n=1..54);
# 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-2), 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-2], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)
PROG
(PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k) * prod( j=1, k-1, 1 + x^(2*j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))} /* Michael Somos, Jan 26 2008 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 27 2006
STATUS
approved