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A327554 Number of partitions in all twice partitions of n where the second partition is strict.

%I A327554 #16 Dec 18 2020 04:00:25
%S A327554 0,1,3,7,15,29,60,108,201,364,643,1106,1944,3253,5493,9183,15161,
%T A327554 24727,40559,65173,104963,167747,266452,420329,663658,1036765,1618221,
%U A327554 2514169,3891121,5992868,9224213,14107699,21548428,32798065,49779331,75301296,113757367
%N A327554 Number of partitions in all twice partitions of n where the second partition is strict.
%H A327554 Alois P. Heinz, <a href="/A327554/b327554.txt">Table of n, a(n) for n = 0..4000</a>
%e A327554 a(3) = 7 = 1+1+2+3 counting the partitions in 3, 21, 2|1, 1|1|1.
%p A327554 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A327554      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A327554     end:
%p A327554 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A327554       b(n, i-1) +(p-> p+[0, p[1]])(g(i)*b(n-i, min(n-i, i)))))
%p A327554     end:
%p A327554 a:= n-> b(n$2)[2]:
%p A327554 seq(a(n), n=0..42);
%t A327554 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 A327554 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, b[n, i - 1] + Function[p, p + {0, p[[1]]}][g[i] b[n - i, Min[n - i, i]]]]];
%t A327554 a[n_] := b[n, n][[2]];
%t A327554 a /@ Range[0, 42] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327554 Cf. A000009, A000041, A270995, A327608.
%K A327554 nonn
%O A327554 0,3
%A A327554 _Alois P. Heinz_, Sep 16 2019