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A090858
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Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.
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24
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
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)
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
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STATUS
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approved
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