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A327607
Number of parts in all twice partitions of n where the first partition is strict.
6
0, 1, 3, 11, 21, 58, 128, 276, 516, 1169, 2227, 4324, 8335, 15574, 29116, 55048, 97698, 176291, 323277, 563453, 1005089, 1770789, 3076868, 5293907, 9184885, 15668638, 26751095, 45517048, 76882920, 128738414, 217219751, 360525590, 599158211, 995474365
OFFSET
0,3
LINKS
EXAMPLE
a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.
MAPLE
g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
numtheory[tau](j), j=1..n)])(combinat[numbpart])
end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, [1, 0], b(n, i-1) +(h-> (f-> f+[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
MATHEMATICA
g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 18 2019
STATUS
approved