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

0,3

Links

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

Example

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.

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 *)

Crossrefs

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

Column k=2 of A327618.

Sequence in context: A120901 A222988 A349222 * A034530 A125246 A302762

Adjacent sequences: A327591 A327592 A327593 * A327595 A327596 A327597

Keyword

nonn

Author

Alois P. Heinz, Sep 18 2019

Status

approved