A115626 Number of non-squashing partitions of {1,...,n}.

1, 1, 2, 4, 14, 26, 107, 190, 1234, 2182, 9947, 17414, 126953, 228398, 1039404, 1857419, 19047146, 35215110, 168364007, 307674658, 2378963269, 4429446046, 20237375204, 37371654467, 410117798653, 776233491226, 3797821367602

Offset: 0

Author

Christian G. Bower, Jan 26 2006

Keywords

• nonn

Comments

A "non-squashing" partition of n is one where n=p_1+p_2+...+p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k.

Links

• N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
• N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Crossrefs

Cf. A018819, A115625.

Programs

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n, i]*a[i], {i, 0, Ceiling[ n/2] - 1}] + If[EvenQ[n], Binomial[n, n/2] (a[n/2] - 1/2), 0];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 10 2018 *)

Formula

a(n) = Sum_{i = 0..ceiling(n/2)-1} (binomial(n, i)*a(i)) + [if n is even] binomial(n, n/2)*(a(n/2)-1/2).