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A327553 Number of partitions in all twice partitions of n where both partitions are strict.

%I A327553 #16 Dec 18 2020 04:00:17
%S A327553 0,1,1,4,6,11,20,33,57,100,165,254,417,649,1039,1648,2540,3836,6020,
%T A327553 9035,13645,20752,31054,45993,68668,101511,149525,220132,321614,
%U A327553 468031,684124,989703,1427054,2064859,2964987,4254028,6090453,8686574,12366583,17598885
%N A327553 Number of partitions in all twice partitions of n where both partitions are strict.
%H A327553 Alois P. Heinz, <a href="/A327553/b327553.txt">Table of n, a(n) for n = 0..5000</a>
%e A327553 a(3) = 4 = 1+1+2 counting the partitions in 3, 21, 2|1.
%p A327553 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A327553      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A327553     end:
%p A327553 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A327553      `if`(n=0, [1, 0], b(n, i-1)+(p-> p+[0, p[1]])(
%p A327553        g(i)*b(n-i, min(n-i, i-1)))))
%p A327553     end:
%p A327553 a:= n-> b(n$2)[2]:
%p A327553 seq(a(n), n=0..42);
%t A327553 g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
%t A327553 b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, {0, 0}, If[n==0, {1, 0}, b[n, i-1] + Function[p, p + {0, p[[1]]}][g[i] b[n-i, Min[n-i, i-1]]]]];
%t A327553 a[n_] := b[n, n][[2]];
%t A327553 a /@ Range[0, 42] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327553 Cf. A000009, A279785, A327605.
%K A327553 nonn
%O A327553 0,4
%A A327553 _Alois P. Heinz_, Sep 16 2019