A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

0, 0, 0, 1, 1, 2, 5, 3, 9, 11, 10, 24, 21, 30, 50, 43, 75, 93, 96, 161, 170, 215, 312, 323, 456, 574, 639, 906, 1046, 1276, 1710, 1935, 2501, 3135, 3642, 4760, 5699, 6893, 8823, 10401, 12952, 16079, 19104, 24002, 29097, 35165, 43865, 52628, 64503, 79363, 95329

Offset

0,6

Comments

A rise is any pair of parts (p_{i-1},p_i) with p_{i-1} < p_i.

By reversal a(n) is also the number of descents in all compositions of n of this kind.

Links

Table of n, a(n) for n=0..50.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,2,-1,0,-2,0,-1).

Formula

G.f.: x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2.

Example

For n = 6 the following compositions have 5 rises: (1,2,1,2), (1,2,3), (2,1,2,1), (3,2,1).

Prog

(PARI)
A_x(N) = {my(x='x+O('x^N)); concat([0, 0, 0], Vec(x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2))}
A_x(40)

Crossrefs

Cf. A011782, A076118, A124760, A151842, A173258, A214247, A220062, A238343.

Sequence in context: A119435 A352783 A193972 * A361398 A332357 A305126

Adjacent sequences: A383546 A383547 A383548 * A383550 A383551 A383552

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 29 2025

Status

approved