A243149 Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.

1, 1, 0, 0, 4, 3, 4, 0, 11, 31, 70, 177, 242, 382, 482, 874, 1655, 4440, 10696, 24390, 49867, 95850, 172980, 289229, 492233, 811753, 1468084, 2813206, 5929361, 12780690, 27858421, 59275097, 122326098, 243179349, 467856049, 873044584, 1588187110, 2842593612

Offset: 0

Author

Alois P. Heinz, May 30 2014

Keywords

• nonn

Examples

			a(8) = 11: [1,1,3,3], [1,3,1,3], [1,3,3,1], [3,1,1,3], [3,1,3,1], [3,3,1,1], [1,1,1,1,4], [1,1,1,4,1], [1,1,4,1,1], [1,4,1,1,1], [4,1,1,1,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A114638 (the same for partitions).

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!,
      `if`(i<1, 0, expand(add(x^`if`(j=0, 0, i-j)*
       b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
    end:
a:= n-> coeff(b(n$2, 0), x, 0):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, Expand[Sum[x^If[j == 0, 0, i - j]*b[n - i*j, i - 1, p + j]/j!, {j,0, n/i}]]]];
a[n_] := Coefficient[b[n, n, 0], x, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 21 2018, translated from Maple *)