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The decapitation of such a partition (delete the greatest part) is term-wise less than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

The number of binary strings sharing the same autocorrelations.

Appears to be row sums of A155092, beginning from a(2). - Mats Granvik, Jan 20 2009

The number of binary words of length n (beginning with 0) which do not start with an even palindrome (i.e. which are not of the form ss*t where s is a (nonempty) word, s* is its reverse, and t is any (possibly empty) word). - Mamuka Jibladze, Sep 30 2014

From Gus Wiseman, Mar 08 2021: (Start)

This sequence counts each of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n in which all adjacent elements have quotients > 1/2. For example, the a(1) = 1 through a(6) = 10 sets are:

{1} {2} {3} {4} {5} {6}

{2,3} {3,4} {3,5} {4,6}

{2,3,4} {4,5} {5,6}

{2,3,5} {3,4,6}

{3,4,5} {3,5,6}

{2,3,4,5} {4,5,6}

{2,3,4,6}

{2,3,5,6}

{3,4,5,6}

{2,3,4,5,6}

2. For n > 1, sets of distinct positive integers with maximum n - 1 whose first-differences are term-wise less than their decapitation (remove the maximum). For example, the set q = {2,4,5} has first-differences (2,1), which are not less than (2,4), so q is not counted under a(5). On the other hand, r = {2,3,5,6} has first-differences {1,2,1}, which are less than {2,3,5}, so r is counted under a(6).

3. Compositions of n where each part after the first is less than the sum of all preceding parts. For example, the a(1) = 1 through a(6) = 10 compositions are:

(1) (2) (3) (4) (5) (6)

(21) (31) (41) (51)

(211) (32) (42)

(311) (411)

(212) (321)

(2111) (312)

(3111)

(2121)

(2112)

(21111)

(End)

Apparently a(n) = number of partitions (p_1, p_2, ..., p_k) of n+1, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i > p_{i+1}+...+p_k. - John McKay (mac(AT)mathstat.concordia.ca), Mar 06 2009

Comment from John McKay confirmed in paper by Bessenrodt, Olsson, and Sellers. Such partitions are called "strongly decreasing" partitions in the paper, see the function s(n) therein.

Also the number of unlabeled binary rooted trees with 2*n + 3 nodes in which the two branches directly under any given non-leaf node are either equal or at least one of them is a leaf. - Gus Wiseman, Oct 08 2018

From Gus Wiseman, Apr 06 2021: (Start)

This sequence counts both of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n + 1 in which all adjacent elements have quotients < 1/2. For example, the a(0) = 1 through a(8) = 7 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8} {9}

{1,3} {1,4} {1,5} {1,6} {1,7} {1,8} {1,9}

{2,5} {2,6} {2,7} {2,8} {2,9}

{3,7} {3,8} {3,9}

{1,3,7} {1,3,8} {4,9}

{1,3,9}

{1,4,9}

2. Sets of distinct positive integers with maximum n + 1 whose first differences are term-wise greater than their decapitation (remove the maximum). For example, the set q = {1,4,9} has first differences (3,5), which are greater than (1,4), so q is counted under a(8). On the other hand, r = {1,5,9} has first differences (4,4), which are not greater than (1,5), so r is not counted under a(8).

Also the number of partitions of n + 1 into powers of 2 covering an initial interval of powers of 2. For example, the a(0) = 1 through a(8) = 7 partitions are:

1 11 21 211 221 2211 421 4211 4221

111 1111 2111 21111 2221 22211 22221

11111 111111 22111 221111 42111

211111 2111111 222111

1111111 11111111 2211111

21111111

111111111

(End)

Also Heinz numbers of integer partitions with no adjacent parts having quotient > 2 (counted by A342094). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).

Also the number of factorizations of n where each factor is greater than the product of all previous factors.

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).

Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

The triangular array (g(n,k)) at A274190 is defined as follows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1.

From Gus Wiseman, Mar 12 2021: (Start)

Also (apparently) the number of compositions of n where all adjacent parts (x, y), satisfy x < 2y. For example, the a(1) = 1 through a(6) = 12 compositions are:

(1) (2) (3) (4) (5) (6)

(11) (12) (13) (14) (15)

(111) (22) (23) (24)

(112) (32) (33)

(1111) (113) (114)

(122) (123)

(1112) (132)

(11111) (222)

(1113)

(1122)

(11112)

(111111)

(End)