cp's OEIS Frontend

Also a(n) = 2nd elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).

Also a(n) = one half the sum of the heights, over all Dyck (n+2)-paths, of the vertices that are at even height and terminate an upstep. For example with n=1, these vertices are indicated by asterisks in the 5 Dyck 3-paths: UU*UDDD, UU*DU*DD, UDUU*DD, UDUDUD, UU*DDUD, yielding a(1)=(2+4+2+0+2)/2=5. - David Callan, Jul 14 2006

Hankel transform is (-1)^n*(2n+1); the Hankel transform of sum(k=0..n, C(2*n,k) ) - C(2n,n) is (-1)^n*n. - Paul Barry, Jan 21 2007

Row sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2) (A187926). - Emanuele Munarini, Mar 16 2011

From Gus Wiseman, Jul 19 2021: (Start)

For n > 0, a(n-1) is also the number of integer compositions of 2n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A053754 /\ A345921. For example, the a(3-1) = 22 compositions of 6 are:

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

(2,4) (1,2,3) (1,1,3,1) (1,1,2,1,1)

(4,2) (1,4,1) (1,2,1,2) (2,1,1,1,1)

(5,1) (2,1,3) (1,3,1,1)

(2,2,2) (2,1,2,1)

(3,1,2) (3,1,1,1)

(3,2,1)

(4,1,1)

(End)

Inverse binomial transform of A027914. Hankel transform (see A001906 for definition) is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 21 2005

Number of walks of length n on a line that starts at the origin and ends at or above 0. - Benjamin Phillabaum, Mar 05 2011

Number of binary integers (i.e., with a leading 1 bit) of length n+1 which have a majority of 1-bits. E.g., for n+1=4: (1011, 1101, 1110, 1111) a(3)=4. - Toby Gottfried, Dec 11 2011

Number of distinct symmetric staircase walks connecting opposite corners of a square grid of side n > 1. - Christian Barrientos, Nov 25 2018

From Gus Wiseman, Aug 20 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. These compositions are ranked by A345917. For example, the a(0) = 1 through a(4) = 11 compositions are:

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

(21) (31) (32)

(111) (112) (41)

(211) (113)

(122)

(212)

(221)

(311)

(1121)

(2111)

(11111)

The following relate to these compositions:

- The unordered version is A027193.

- The complement is counted by A058622.

- The reverse unordered version is A086543.

- The version for alternating sum >= 0 is A116406.

- The version for alternating sum < 0 is A294175.

- Ranked by A345917. (End)

The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

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

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.

a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006

Number of partitions of n where all parts (except for possibly the first part) are even; see example. - Joerg Arndt, Apr 22 2013

For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p. For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Oct 28 2021: (Start)

If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.

Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:

11 111 22 221 33 331 44 441 55

1111 11111 2211 22111 2222 22221 3322

111111 1111111 3311 33111 4411

221111 2211111 222211

11111111 111111111 331111

22111111

1111111111

These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.

(End)

Also the number of integer partitions of n that are of length 2 or contain n/2. The first condition alone is A004526, complement A058984. The second condition alone is A035363, complement A086543, ranks A344415. - Gus Wiseman, Oct 07 2023

Row sums = A000041, the partition numbers.

The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.

k-th triangle in the infinite set can be defined as having the sequence:

"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].

Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.

Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.

All triangles in the infinite set have row sums = A000041.

The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:

...

k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies

.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =

.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...

...

k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies

.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get

.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...

...

k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies

.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get

.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...

...

Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

All such partitions have odd length.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

From Gus Wiseman, Jan 21 2022: (Start)

Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:

1 11 21 22 32 33 43 44 54

111 1111 221 2211 331 2222 441

2111 111111 2221 3311 3222

11111 3211 221111 3321

22111 11111111 4311

211111 22221

1111111 33111

222111

321111

2211111

21111111

111111111

(End)