A258473 Number of partitions of n into two sorts of parts having exactly 3 parts of the second sort.

1, 5, 16, 41, 91, 186, 351, 635, 1090, 1824, 2939, 4652, 7162, 10875, 16159, 23758, 34321, 49145, 69389, 97213, 134608, 185172, 252182, 341443, 458413, 612186, 811567, 1070826, 1403784, 1832370, 2378320, 3074642, 3954869, 5068684, 6466697, 8222640, 10412903

Offset: 3

Author

Alois P. Heinz, May 31 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A256193.

Programs

Maple

b:= proc(n, i) option remember; series(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
       binomial(j, t), t=0..min(3, j)), j=0..n/i))), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..40);

Mathematica

b[n_, i_] := b[n, i] = Series[If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*Sum[ x^t*Binomial[j, t], {t, 0, Min[3, j]}], {j, 0, n/i}]]], {x, 0, 4}];
a[n_] := Coefficient[b[n, n], x, 3];
a /@ Range[3, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)