OFFSET
0,5
COMMENTS
Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.
Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
FORMULA
G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018
EXAMPLE
a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From Gus Wiseman, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
(11) (22) (221) (33) (322) (44) (441) (55) (443)
(211) (311) (411) (331) (332) (522) (433) (533)
(511) (422) (711) (442) (551)
(3211) (611) (3321) (622) (722)
(3221) (4221) (811) (911)
(4211) (4311) (5221) (4322)
(5211) (5311) (4331)
(6211) (4421)
(5411)
(6221)
(6311)
(7211)
(43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
(2) (22) (32) (222) (322) (332) (432) (3322)
(31) (311) (3111) (331) (431) (3222) (3331)
(421) (2222) (4221) (22222)
(31111) (3311) (4311) (42211)
(4211) (33111) (43111)
(311111) (42111) (331111)
(3111111) (421111)
(31111111)
(End)
MAPLE
g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i, i=1..k), k=1..15): gser:=series(g, x=0, 64): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
`if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
`if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100); # Alois P. Heinz, Dec 28 2015
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Length[#]-Length[Union[#]]==1&]], {n, 0, 30}] (* Gus Wiseman, Apr 19 2019 *)
PROG
(PARI) alist(n)=concat([0, 0], Vec(sum(k=1, n\2, (x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1, n-2*k, 1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Feb 12 2004
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
a(0) added by Franklin T. Adams-Watters, Nov 02 2015
STATUS
approved