OFFSET
0,6
COMMENTS
Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Aug 06 2004
From Gus Wiseman, Sep 27 2025: (Start)
Also the number of superdiagonal strict partitions of n, where a sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i = 1..k. For example, the a(10) = 4 through a(15) = 10 superdiagonal strict partitions are:
(10) (11) (12) (13) (14) (15)
(6,4) (6,5) (7,5) (7,6) (8,6) (8,7)
(7,3) (7,4) (8,4) (8,5) (9,5) (9,6)
(8,2) (8,3) (9,3) (9,4) (10,4) (10,5)
(9,2) (10,2) (10,3) (11,3) (11,4)
(5,4,3) (11,2) (12,2) (12,3)
(6,4,3) (6,5,3) (13,2)
(7,4,3) (6,5,4)
(7,5,3)
(8,4,3)
The strict complement is counted by A237976.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
I. Martinjak and D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
FORMULA
G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - Joerg Arndt, Jan 29 2011
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - Vaclav Kotesovec, Jan 02 2016
EXAMPLE
a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.
From Gus Wiseman, Sep 27 2025: (Start)
The a(10) = 4 through a(15) = 10 partitions are the following. Their Heinz numbers are given by A325162.
(10) (11) (12) (13) (14) (15)
(7,3) (7,4) (8,4) (8,5) (9,5) (9,6)
(8,2) (8,3) (9,3) (9,4) (10,4) (10,5)
(9,1) (9,2) (10,2) (10,3) (11,3) (11,4)
(10,1) (11,1) (11,2) (12,2) (12,3)
(7,4,1) (12,1) (13,1) (13,2)
(8,4,1) (8,5,1) (14,1)
(9,4,1) (8,5,2)
(9,5,1)
(10,4,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>
ceil(i*(i+3)/6), 0, b(n, i-1)+b(n-i, min(n-i, i-3))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..70); # Alois P. Heinz, Jan 26 2022
MATHEMATICA
nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 21 2013 *)
suppQ[mset_]:=And@@Table[mset[[i]]>=i, {i, Length[mset]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&suppQ[#]&]], {n, 30}] (* Gus Wiseman, Sep 27 2025 *)
PROG
(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 13 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Prepended a(0)=1, Joerg Arndt, Jul 21 2013
STATUS
approved