A327594 Number of parts in all twice partitions of n.

0, 1, 5, 14, 44, 100, 274, 581, 1417, 2978, 6660, 13510, 29479, 58087, 120478, 236850, 476913, 916940, 1812498, 3437043, 6657656, 12512273, 23780682, 44194499, 83117200, 152837210, 283431014, 517571202, 949844843, 1719175176, 3127751062, 5618969956, 10133425489

Offset: 0

Author

Alois P. Heinz, Sep 18 2019

Keywords

• nonn

Examples

			a(2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
a(3) = 14 = 1+2+3+2+3+3: 3, 21, 111, 2|1, 11|1, 1|1|1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3200

Crossrefs

Cf. A000041, A006128, A063834, A327590, A327605, A327607, A327608.

Column k=2 of A327618.

Programs

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`(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)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
         (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
    end:
a:= n-> b(n$2, 2)[2]:
seq(a(n), n=0..37);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)