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A209815 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 A209815 #22 Apr 18 2019 03:23:12
%S A209815 0,1,4,10,23,47,90,164,288,488,807,1303,2063,3210,4920,7434,11098,
%T A209815 16380,23928,34624,49668,70667,99795,139935,194930,269857,371413,
%U A209815 508363,692195,937838,1264685,1697810,2269557,3021462,4006812,5293650,6968730,9142306,11954194
%N A209815 Number of partitions of 2n in which every part is <n; also, the number of partitions of 2 into rational numbers a/b such that 0<a<b<=n and b divides n.
%H A209815 Alois P. Heinz, <a href="/A209815/b209815.txt">Table of n, a(n) for n = 1..1000</a>
%F A209815 a(n) = A008284(3*n-1,n-1). - _Hans Loeblich_ Apr 18 2019
%e A209815 The 4 partitions of 6 with parts <3:
%e A209815 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1.
%e A209815 Matching partitions of 2 into rationals as described:
%e A209815 2/3 + 2/3 + 2/3
%e A209815 2/3 + 2/3 + 1/3 + 1/3
%e A209815 2/3 + 1/3 + 1/3 + 1/3 + 1/3
%e A209815 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.
%p A209815 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A209815       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A209815     end:
%p A209815 a:= n-> b(2*n, n-1):
%p A209815 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2012
%t A209815 f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n - 1 &]];  Table[f[n], {n, 1, 34}]  (* A209815 *)
%t A209815 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-1]; Table [a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 28 2015, after _Alois P. Heinz_ *)
%o A209815 (Haskell)
%o A209815 a209815 n = p [1..n-1] (2*n) where
%o A209815    p _          0 = 1
%o A209815    p []         _ = 0
%o A209815    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A209815 -- _Reinhard Zumkeller_, Nov 14 2013
%Y A209815 Cf. A209816.
%Y A209815 Cf. A231429.
%K A209815 nonn
%O A209815 1,3
%A A209815 _Clark Kimberling_, Mar 13 2012
%E A209815 More terms from _Alois P. Heinz_, Jul 09 2012