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A327554
Number of partitions in all twice partitions of n where the second partition is strict.
4
0, 1, 3, 7, 15, 29, 60, 108, 201, 364, 643, 1106, 1944, 3253, 5493, 9183, 15161, 24727, 40559, 65173, 104963, 167747, 266452, 420329, 663658, 1036765, 1618221, 2514169, 3891121, 5992868, 9224213, 14107699, 21548428, 32798065, 49779331, 75301296, 113757367
OFFSET
0,3
LINKS
EXAMPLE
a(3) = 7 = 1+1+2+3 counting the partitions in 3, 21, 2|1, 1|1|1.
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
b(n, i-1) +(p-> p+[0, p[1]])(g(i)*b(n-i, min(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);
MATHEMATICA
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];
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]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 16 2019
STATUS
approved