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%I #31 Sep 14 2023 08:37:07
%S 1,2,3,5,7,10,13,17,20,26,29,35,39,48,48,60,61,74,73,87,86,106,99,120,
%T 112,140,130,155,143,176,159,194,180,216,186,240,209,258,234,274,243,
%U 308,261,325,289,348,297,383,314,392,356,423,355,460,372,468,422
%N Number of partitions of n having no parts strictly between the smallest and the largest part (n>=1).
%H Alois P. Heinz, <a href="/A265250/b265250.txt">Table of n, a(n) for n = 1..10000</a>
%H Jonathan Bloom, Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.
%F a(n) = A265249(n,0).
%F G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/ ((1-x^i)*(1-x^j)).
%F a(n) = A116608(n,1) + A116608(n,2) = A000005(n) + A002133(n). - _Seiichi Manyama_, Sep 14 2023
%e a(3) = 3 because we have [3], [1,2], [1,1,1] (all partitions of 3).
%e a(6) = 10 because we have all A000041(6) = 11 partitions of 6 except [1,2,3].
%e a(7) = 13 because we have all A000041(7) = 15 partitions of 7 except [1,2,4] and [1,1,2,3].
%p g := add(x^i/(1-x^i), i = 1 .. 80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)), j = i+1..80),i=1..80): gser := series(g,x=0,60): seq(coeff(gser,x,n),n=1..50);
%p # second Maple program:
%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
%p add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
%p end:
%p a:= n-> b(n$2, 2):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 01 2016
%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y Cf. A000041, A265249.
%Y Cf. A000005, A002133, A116608, A309058.
%K nonn,look
%O 1,2
%A _Emeric Deutsch_, Dec 25 2015
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