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A209816 Number of partitions of 2n in which every part is also, the number of partitions of 2 into rational numbers a/b such that 0 and b divides n.

%I A209816 #62 Oct 30 2021 15:04:46
%S A209816 1,3,7,15,30,58,105,186,318,530,863,1380,2164,3345,5096,7665,11395,
%T A209816 16765,24418,35251,50460,71669,101050,141510,196888,272293,374423,
%U A209816 512081,696760,943442,1271527,1706159,2279700,3033772,4021695,5311627,6990367,9168321
%N A209816 Number of partitions of 2n in which every part is <n+1; also, the number of partitions of 2 into rational numbers a/b such that 0<a<=b<=n and b divides n.
%C A209816 Also, the number of partitions of 3n in which n is the maximal part.
%C A209816 Also, the number of partitions of 3n into n parts. - _Seiichi Manyama_, May 07 2018
%C A209816 Also the number of multigraphical partitions of 2n, i.e., integer partitions that comprise the multiset of vertex-degrees of some multigraph. - _Gus Wiseman_, Oct 24 2018
%C A209816 Also number of partitions of 2n with at most n parts. Conjugate partitions map one to one to partitions of 2*n with each part <= n. - _Wolfdieter Lang_, May 21 2019
%H A209816 Alois P. Heinz, <a href="/A209816/b209816.txt">Table of n, a(n) for n = 1..1000</a>
%H A209816 Gus Wiseman, <a href="/A209816/a209816.png">Multigraphs realizing each of the a(4) = 15 multigraphical partitions of 8.</a>
%H A209816 Gus Wiseman, <a href="/A209816/a209816_1.png">Multigraphs realizing each of the a(5) = 30 multigraphical partitions of 10.</a>
%F A209816 a(n) = A000041(2*n)-A000070(n-1). - _Matthew Vandermast_, Jul 16 2012
%F A209816 a(n) = Sum_{k=1..n} A008284(2*n, k) = A000041(2*n) - A000070(n-1), for n >= 1. - _Wolfdieter Lang_, May 21 2019
%e A209816 The 7 partitions of 6 with parts <4 are as follows:
%e A209816 3+3, 3+2+1, 3+1+1+1
%e A209816 2+2+2, 2+2+1+1, 2+1+1+1+1
%e A209816 1+1+1+1+1+1.
%e A209816 Matching partitions of 2 into rationals as described:
%e A209816 1 + 1
%e A209816 1 + 3/3 + 1/3
%e A209816 1 + 1/3 + 1/3 + 1/3
%e A209816 2/3 + 2/3 + 2/3
%e A209816 2/3 + 2/3 + 1/3 + 1/3
%e A209816 2/3 + 1/3 + 1/3 + 1/3 + 1/3
%e A209816 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.
%e A209816 From _Seiichi Manyama_, May 07 2018: (Start)
%e A209816 n | Partitions of 3n into n parts
%e A209816 --+-------------------------------------------------
%e A209816 1 | 3;
%e A209816 2 | 5+1, 4+2, 3+3;
%e A209816 3 | 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3; (End)
%e A209816 From _Gus Wiseman_, Oct 24 2018: (Start)
%e A209816 The a(1) = 1 through a(4) = 15 partitions:
%e A209816   (11)  (22)    (33)      (44)
%e A209816         (211)   (222)     (332)
%e A209816         (1111)  (321)     (422)
%e A209816                 (2211)    (431)
%e A209816                 (3111)    (2222)
%e A209816                 (21111)   (3221)
%e A209816                 (111111)  (3311)
%e A209816                           (4211)
%e A209816                           (22211)
%e A209816                           (32111)
%e A209816                           (41111)
%e A209816                           (221111)
%e A209816                           (311111)
%e A209816                           (2111111)
%e A209816                           (11111111)
%e A209816 (End)
%p A209816 b:= proc(n, i) option remember;
%p A209816       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A209816     end:
%p A209816 a:= n-> b(2*n, n):
%p A209816 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2012
%t A209816 f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n &]]; Table[f[n], {n, 1, 30}] (* A209816 *)
%t A209816 Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,2*n}],{n,1,20}] (* _Vaclav Kotesovec_, May 25 2015 *)
%t A209816 Table[Length@IntegerPartitions[3n, {n}], {n, 25}] (* _Vladimir Reshetnikov_, Jul 24 2016 *)
%t A209816 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%o A209816 (Haskell)
%o A209816 a209816 n = p [1..n] (2*n) where
%o A209816    p _          0 = 1
%o A209816    p []         _ = 0
%o A209816    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A209816 -- _Reinhard Zumkeller_, Nov 14 2013
%Y A209816 Cf. A000041, A000070, A000569, A008284, A025065, A079122, A096373, A147878, A209815, A320911, A320921, A320924.
%K A209816 nonn
%O A209816 1,2
%A A209816 _Clark Kimberling_, Mar 13 2012
%E A209816 More terms from _Alois P. Heinz_, Jul 09 2012