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A327795 Number of parts in all proper twice partitions of n into distinct parts.

%I A327795 #14 Dec 09 2020 15:15:02
%S A327795 0,0,0,3,6,13,30,61,121,210,353,600,989,1628,2667,4205,6514,10406,
%T A327795 15893,24322,37516,56824,85102,128420,191579,284898,422839,622721,
%U A327795 913006,1345320,1958269,2843788,4140170,5983662,8632808,12433730,17830728,25527909,36516161
%N A327795 Number of parts in all proper twice partitions of n into distinct parts.
%H A327795 Alois P. Heinz, <a href="/A327795/b327795.txt">Table of n, a(n) for n = 1..5000</a>
%e A327795 a(4) = 3:
%e A327795   4 -> 31 -> 211   (3 parts)
%p A327795 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327795      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p A327795          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327795         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p A327795     end:
%p A327795 a:= n-> (k-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
%p A327795 seq(a(n), n=1..41);
%t A327795 b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {0, 0}, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][ b[i, i, k - 1]]]]]];
%t A327795 T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t A327795 a[n_] := T[n, 2];
%t A327795 Array[a, 41] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A327795 Column k=2 of A327632.
%Y A327795 Cf. A327605.
%K A327795 nonn
%O A327795 1,4
%A A327795 _Alois P. Heinz_, Sep 25 2019