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A327608
Number of parts in all twice partitions of n where the second partition is strict.
6
0, 1, 3, 8, 17, 34, 74, 134, 254, 470, 842, 1463, 2620, 4416, 7545, 12749, 21244, 34913, 57868, 93583, 151963, 244602, 391206, 620888, 987344, 1550754, 2435087, 3804354, 5920225, 9162852, 14179754, 21785387, 33436490, 51121430, 77935525, 118384318, 179617794
OFFSET
0,3
LINKS
EXAMPLE
a(3) = 8 = 1+2+2+3 counting the parts in 3, 21, 2|1, 1|1|1.
MAPLE
g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, [1, 0], g(n, i-1)+ (f-> f+
[0, f[1]])(g(n-i, min(n-i, i-1)))))
end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i$2)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
MATHEMATICA
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, g[n, i - 1] + Function[f, f + {0, f[[1]]}][g[n - i, Min[n - i, i - 1]]]]];
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 2, 0, b[n, i - 1]] + Module[{h, f}, h = g[i, i]; f = b[n - i, Min[n - i, i]] h[[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