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

0,4

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.

Links

Table of n, a(n) for n=0..34.

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

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

Example

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).

Crossrefs

Cf. A129953.

Sequence in context: A269059 A324218 A229731 * A271904 A272541 A273142

Adjacent sequences: A360948 A360949 A360950 * A360952 A360953 A360954

Keyword

nonn,easy

Author

Enrique Navarrete, Feb 26 2023

Status

approved