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Number of partitions in all twice partitions of n where the second partition is strict.
4

%I #16 Dec 18 2020 04:00:25

%S 0,1,3,7,15,29,60,108,201,364,643,1106,1944,3253,5493,9183,15161,

%T 24727,40559,65173,104963,167747,266452,420329,663658,1036765,1618221,

%U 2514169,3891121,5992868,9224213,14107699,21548428,32798065,49779331,75301296,113757367

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

%H Alois P. Heinz, <a href="/A327554/b327554.txt">Table of n, a(n) for n = 0..4000</a>

%e a(3) = 7 = 1+1+2+3 counting the partitions in 3, 21, 2|1, 1|1|1.

%p g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(

%p `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p end:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p b(n, i-1) +(p-> p+[0, p[1]])(g(i)*b(n-i, min(n-i, i)))))

%p end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..42);

%t 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 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 a[n_] := b[n, n][[2]];

%t a /@ Range[0, 42] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)

%Y Cf. A000009, A000041, A270995, A327608.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 16 2019