A327554 Number of partitions in all twice partitions of n where the second partition is strict.

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

Alois P. Heinz, Table of n, a(n) for n = 0..4000

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

Cf. A000009, A000041, A270995, A327608.

Sequence in context: A283258 A080011 A284022 * A120538 A146019 A284421

Adjacent sequences: A327551 A327552 A327553 * A327555 A327556 A327557

Keyword

nonn

Author

Alois P. Heinz, Sep 16 2019

Status

approved