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

0,5

Links

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

Example

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].

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 *)

Crossrefs

Cf. A114638 (the same for partitions).

Sequence in context: A306769 A336031 A329982 * A048156 A070431 A070511

Adjacent sequences: A243146 A243147 A243148 * A243150 A243151 A243152

Keyword

nonn

Author

Alois P. Heinz, May 30 2014

Status

approved