A360951 Expansion of e.g.f. (cosh(x) - 1)*(1 + x)*exp(x).

0, 0, 1, 6, 19, 50, 121, 280, 631, 1398, 3061, 6644, 14323, 30706, 65521, 139248, 294895, 622574, 1310701, 2752492, 5767147, 12058602, 25165801, 52428776, 109051879, 226492390, 469762021, 973078500, 2013265891, 4160749538, 8589934561, 17716740064, 36507221983, 75161927646, 154618822621

Offset: 0

Author

Enrique Navarrete, Feb 26 2023

Keywords

• nonn
• easy

Comments

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first set has an even number of elements greater than or equal to two, the second set has either one or no elements, and the third set has no restrictions.

Examples

			The 19 set partitions for n=4 are the following:
  {1,2,3,4}, { }, { } (one of these);
  {1,2}, { }, {3,4}   (6 of these);
  {1,2}, {3}, {4}     (12 of these).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A129953.

Formula

a(n) = 2^(n-1) + n*2^(n-2) - n - 1 for n >= 2.

G.f.: x^2*(1 - 4*x^2 + 2*x^3)/((1 - x)^2*(1 - 2*x)^2). - Stefano Spezia, Mar 04 2023

a(n) = (n + 2)*2^(n-2) - n - 1 = A129953(n) - n - 1 for n >= 2. - Stefano Spezia, Mar 05 2023