cp's OEIS Frontend

a(n) is the number of n-digit binary sequences that have more 1's than 0's. - Geoffrey Critzer, Jul 16 2009

Maps to the number of walks that end above 0 on the number line with steps being 1 or -1. - Benjamin Phillabaum, Mar 06 2011

Chris Godsil observes that a(n) is the independence number of the (n+1)-folded cube graph; proof is by a Cvetkovic's eigenvalue bound to establish an upper bound and a direct construction of the independent set by looking at vertices at an odd (resp., even) distance from a fixed vertex when n is odd (resp., even). - Stan Wagon, Jan 29 2013

Also the number of subsets of {1,2,...,n} that contain more odd than even numbers. For example, for n=4, a(4)=5 and the 5 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}. See A014495 when same number of even and odd numbers. - Enrique Navarrete, Feb 10 2018

Also half the number of length-n binary sequences with a different number of zeros than ones. This is also the number of integer compositions of n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Also the number of integer compositions of n+1 with alternating sum <= 0, ranked by A345915 (reverse: A345916). - Gus Wiseman, Jul 19 2021

We define a composition to be alternating including twins (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. Except in the case of twins (x,x), all such compositions are anti-runs (A003242). These compositions avoid the weak consecutive patterns (1,2,3) and (3,2,1), the strict version being A344614.

The version without twins (x,x) is A025047 (alternating compositions).

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2n with alternating sum 2k.

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

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.

These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.

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

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

Also the number of reversed integer partitions of 2n with alternating sum >= 0.

The reverse-alternating sum of a partition is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.

From Gus Wiseman, Jun 08 2021: (Start)

Also the number of:

- integer partitions of 2n with reverse-alternating sum 2;

- reversed integer partitions of 2n with alternating sum 2;

- integer partitions of 2n with exactly two odd parts, one of which is the greatest;

- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.

Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.

(End)

Second binomial transform of A032443 with interpolated zeros.

a(n) is the total number of lattice points, taken over all Dyck n-paths (A000108), that (i) lie on or above ground level and (ii) lie on or directly below a peak. For example with n = 2, UUDD has 1 peak contributing 3 lattice points--(2, 0), (2, 1) and (2, 2) when the path starts at the origin--and UDUD has 2 peaks, each contributing 2 lattice points and so a(2) = 3 + 4 = 7. - David Callan, Jul 14 2006

Hankel transform is binomial(n + 2, 2). - Paul Barry, Dec 04 2007

Image of (-1)^n under the Riordan array ((1/2)(1/(1 - 4x) + 1/sqrt(1 - 4x)), c(x) - 1), c(x) the g.f. of A000108. - Paul Barry, Jun 15 2008

From Gus Wiseman, Jun 21 2021: (Start)

Also the even bisection of A116406 = number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(3) = 26 compositions are:

(6) (33) (114) (1122) (11112) (111111)

(42) (123) (1131) (11121)

(51) (132) (1221) (11211)

(213) (2112) (12111)

(222) (2121) (21111)

(231) (2211)

(312) (3111)

(321)

(411)

(End)

Number of subsets of {1,2,...,n} that contain more even than odd numbers.

Note that A058622 counts the nonempty subsets of {1,2,...,n} that contain more odd than even numbers.

From Gus Wiseman, Jul 22 2021: (Start)

Also the number of integer compositions of n + 1 with alternating sum < 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(0) = 0 through a(6) = 6 compositions (empty columns indicated by dots) are:

. . (12) (13) (14) (15)

(23) (24)

(131) (141)

(1112) (1113)

(1211) (1212)

(1311)

Also the number of integer compositions of n + 1 with reverse-alternating sum < 0. For a bijection, keep the odd-length compositions and reverse the even-length ones.

Also the number of (n+1)-digit binary numbers with more 0's than 1's. For example, the a(0) = 0 through a(5) = 6 binary numbers are:

. . 100 1000 10000 100000

10001 100001

10010 100010

10100 100100

11000 101000

110000

(End)

2*a(n) is the number of all-positive pinnacle sets that are admissible in the group S_{n+1}^B of signed permutations, but not admissible in S_{n+1}. - Bridget Tenner, Jan 06 2023