A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

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

Cf. A047967, A265251.

Column k=2 of A266477.

Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

Sequence in context: A336032 A093393 A341095 * A187504 A036654 A262669

Adjacent sequences: A090855 A090856 A090857 * A090859 A090860 A090861

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