cp's OEIS Frontend

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

Table starts

...4.....16.......36.........64.........100.........144..........196

...0.....20.......52........144.........208.........436..........532

..12....120......548.......1504........3292........6376........10564

...0....300.....1632.......7092.......16484.......43440........75080

..40...1284....12692......58824......193232......521124......1142180

...0...4132....51196.....368588.....1399640.....4875112.....11953848

.140..16272...355396....2880240....14715004....55994544....168083116

...0..57808..1657632...20265640...123729664...591604824...2026547348

.504.223308.10858368..156028036..1247614580..6764014136..27843005992

...0.828456.54754656.1154193268.11199296500.75116513672.355600251460

From Gus Wiseman, Oct 12 2022: (Start)

Also the number of integer partitions of n whose skew-alternating sum is 0, where we define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ... These are the conjugates of the partitions described in the name. For example, the a(0) = 1 through a(8) = 10 partitions are:

() . (11) . (22) . (33) . (44)

(211) (321) (422)

(1111) (2211) (431)

(111111) (2222)

(3221)

(3311)

(22211)

(221111)

(2111111)

(11111111)

The ordered version (compositions) is A138364

These partitions are ranked by A357636, reverse A357632.

The reverse version is A357640 (aerated).

Cf. A357623, A357629, A357630, A357634, A357646, A357705.

(End)

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k). For comparison:

A001006(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is even],

A005717(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is odd].

Thus one might simply say: The extended Motzkin numbers are the binomial sum of the extended Catalan numbers. Moreover: The Catalan numbers aerated with 0's at odd positions (A126120) are the inverse binomial transform of the Motzkin numbers (A001006). The complementary Catalan numbers (A001700) aerated with 0's at even positions (A138364) are the inverse binomial transform of the complementary Motzkin numbers (A005717). The extended Catalan numbers (A057977 = A126120 + A138364) are the inverse binomial transform of the extended Motzkin numbers (A189912).

David Scambler observed that [1, a(n-1)] for n >= 1 count the Dyck paths of semilength n which satisfy the condition "number of peaks <= number of returns + number of hills". - Peter Luschny, Oct 22 2012

We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) * i * y_i.

Strings of length 2n+2 over the alphabet {U, R} with n Rs and avoiding UU or RRR as substrings.

Also number of binary words with 3 1's and n 0's that do not contain the substring 101. a(2) = 5: 00111, 10011, 11001, 11100, 01110. - Alois P. Heinz, Jul 18 2013

Let (b(n)) be the p-INVERT of A010892 using p(S) = 1 - S^2; then b(n) = a(n+1) for n >= 0. See A292301. - Clark Kimberling, Sep 30 2017

From Gus Wiseman, Oct 13 2022: (Start)

Also the number of integer compositions of n+3 with half-alternating sum n-1, where we define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ... For example, the a(1) = 2 through a(4) = 10 compositions are:

(112) (122) (132)

(1111) (212) (222)

(1211) (312)

(2111) (1311)

(11111) (2211)

(3111)

(11112)

(12111)

(21111)

(111111)

A001700/A138364 = compositions with alternating sum 0, ranked by A344619.

A357621 = half-alternating sum of standard compositions, reverse A357622.

A357641 = compositions with half-alternating sum 0, ranked by A357625.

Cf. A357136, A357182, A357626, A357631, A357639, A357642, A357706.

(End)

From Gus Wiseman, Jun 16 2023: (Start)

Also appears to be the number of integer compositions of n + 2 with weighted sum greater than reverse-weighted sum, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i, and the reverse is Sum_{i=1..k} i * y_{k-i+1}. The a(1) = 1 through a(4) = 12 compositions are:

(21) (31) (32) (42)

(211) (41) (51)

(221) (231)

(311) (312)

(1211) (321)

(2111) (411)

(1311)

(2121)

(2211)

(3111)

(12111)

(21111)

The version for partitions is A144300, strict A111133.

(End)

First differs from A347050 at a(216) = 28, A347050(216) = 27.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.