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A327594 Number of parts in all twice partitions of n.

%I A327594 #30 Dec 05 2020 07:52:18
%S A327594 0,1,5,14,44,100,274,581,1417,2978,6660,13510,29479,58087,120478,
%T A327594 236850,476913,916940,1812498,3437043,6657656,12512273,23780682,
%U A327594 44194499,83117200,152837210,283431014,517571202,949844843,1719175176,3127751062,5618969956,10133425489
%N A327594 Number of parts in all twice partitions of n.
%H A327594 Alois P. Heinz, <a href="/A327594/b327594.txt">Table of n, a(n) for n = 0..3200</a>
%e A327594 a(2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
%e A327594 a(3) = 14 = 1+2+3+2+3+3: 3, 21, 111, 2|1, 11|1, 1|1|1.
%p A327594 g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
%p A327594       numtheory[tau](j), j=1..n)])(combinat[numbpart])
%p A327594     end:
%p A327594 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A327594       `if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
%p A327594        h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i)))
%p A327594     end:
%p A327594 a:= n-> b(n$2)[2]:
%p A327594 seq(a(n), n=0..37);
%p A327594 # second Maple program:
%p A327594 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327594      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p A327594          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327594         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p A327594     end:
%p A327594 a:= n-> b(n$2, 2)[2]:
%p A327594 seq(a(n), n=0..37);
%t A327594 b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 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], k]]][b[i, i, k - 1]]]];
%t A327594 a[n_] := b[n, n, 2][[2]];
%t A327594 a /@ Range[0, 37] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y A327594 Cf. A000041, A006128, A063834, A327590, A327605, A327607, A327608.
%Y A327594 Column k=2 of A327618.
%K A327594 nonn
%O A327594 0,3
%A A327594 _Alois P. Heinz_, Sep 18 2019