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%I #36 Apr 21 2019 07:44:57
%S 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,
%T 221,264,309,366,433,505,593,696,805,941,1090,1258,1458,1684,1933,
%U 2225,2555,2922,3346,3823,4349,4961,5644,6402,7267,8234,9309,10525,11886,13393
%N Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.
%C 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.
%C 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
%H Alois P. Heinz, <a href="/A090858/b090858.txt">Table of n, a(n) for n = 0..10000</a>
%F 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).
%F 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
%F 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
%e a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
%e From _Gus Wiseman_, Apr 19 2019: (Start)
%e 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.
%e (11) (22) (221) (33) (322) (44) (441) (55) (443)
%e (211) (311) (411) (331) (332) (522) (433) (533)
%e (511) (422) (711) (442) (551)
%e (3211) (611) (3321) (622) (722)
%e (3221) (4221) (811) (911)
%e (4211) (4311) (5221) (4322)
%e (5211) (5311) (4331)
%e (6211) (4421)
%e (5411)
%e (6221)
%e (6311)
%e (7211)
%e (43211)
%e 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.
%e (2) (22) (32) (222) (322) (332) (432) (3322)
%e (31) (311) (3111) (331) (431) (3222) (3331)
%e (421) (2222) (4221) (22222)
%e (31111) (3311) (4311) (42211)
%e (4211) (33111) (43111)
%e (311111) (42111) (331111)
%e (3111111) (421111)
%e (31111111)
%e (End)
%p 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
%p # second Maple program:
%p b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
%p `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
%p `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Dec 28 2015
%t 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_ *)
%t Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* _Gus Wiseman_, Apr 19 2019 *)
%o (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
%Y Cf. A047967, A265251.
%Y Column k=2 of A266477.
%Y Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.
%K easy,nonn
%O 0,5
%A _Vladeta Jovovic_, Feb 12 2004
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
%E a(0) added by _Franklin T. Adams-Watters_, Nov 02 2015
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