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)

a(n) = number of permutations in S_{n+2} containing exactly one 312 pattern. E.g., S_3 has a_1 = 1 permutations containing exactly one 312 pattern, and S_4 has a_2 = 5 permutations containing exactly one 312 pattern, namely 1423, 2413, 3124, 3142, and 4231. This comment is also true if 312 is replaced by any of 132, 213, or 231 (but not 123 or 321, for which see A003517). [Comment revised by N. J. A. Sloane, Nov 26 2022]

Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch, Dec 05 2003

Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch, Dec 05 2003

Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch, Dec 05 2003

Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch, May 20 2004

Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004

Number of jumps in all full binary trees with n+1 internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 18 2007

a(n) is the total number of nonempty Dyck subpaths in all Dyck paths (A000108) of semilength n. For example, the Dyck path UUDUUDDD has Dyck subpaths stretching over positions 1-8 (the entire path), 2-3, 2-7, 4-7, 5-6 and so contributes 5 to a(4). - David Callan, Jul 25 2008

a(n+1) is the total number of ascents in the set of all n-permutations avoiding the pattern 132. For example, a(2) = 5 because there are 5 ascents in the set 123, 213, 231, 312, 321. - Cheyne Homberger, Oct 25 2013

Number of increasing tableaux of shape (n+1,n+1) with largest entry 2n+1. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 5 counts the five tableaux (124)(235), (123)(245), (124)(345), (134)(245), (123)(245). - Oliver Pechenik, May 02 2014

a(n) is the number of noncrossing partitions of 2n+1 into n-1 blocks of size 2 and 1 block of size 3. - Oliver Pechenik, May 02 2014

Number of paths in the half-plane x>=0, from (0,0) to (2n+1,3), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 5 paths: UUUUD, UUUDU, UUDUU, UDUUU, DUUUU. - José Luis Ramírez Ramírez, Apr 19 2015

From Gus Wiseman, Aug 20 2021: (Start)

Also the number of binary numbers with 2n+2 digits and with two more 0's than 1's. For example, the a(2) = 5 binary numbers are: 100001, 100010, 100100, 101000, 110000, with decimal values 33, 34, 36, 40, 48. Allowing first digit 0 gives A001791, ranked by A345910/A345912.

Also the number of integer compositions of 2n+2 with alternating sum -2, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 21 compositions are:

(35) (152) (1124) (11141) (111113)

(251) (1223) (12131) (111212)

(1322) (13121) (111311)

(1421) (14111) (121112)

(2114) (121211)

(2213) (131111)

(2312)

(2411)

The following pertain to these compositions:

- The unordered version is A344741.

- Ranked by A345924 (reverse: A345923).

- A345197 counts compositions by length and alternating sum.

- A345925 ranks compositions with alternating sum 2 (reverse: A345922).

(End)

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004

a(n) is the 2n-th moment of the distance from the origin of a 3-step random walk in the plane. - Peter M. W. Gill (peter.gill(AT)nott.ac.uk), Feb 27 2004

a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. - Jeffrey Shallit, Aug 17 2010

Consider 2D simple random walk on honeycomb lattice. a(n) gives number of paths of length 2n ending at origin. - Sergey Perepechko, Feb 16 2011

Row sums of A318397 the square of A008459. - Peter Bala, Mar 05 2013

Conjecture: For each n=1,2,3,... the polynomial g_n(x) = Sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013

This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

a(n) is the sum of the squares of the coefficients of (x + y + z)^n. - Michael Somos, Aug 25 2018

a(n) is the constant term in the expansion of (1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 28 2019

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).

Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).

Guide to p-INVERT sequences using p(S) = 1 - S - S^2:

t(A000012) = t(1,1,1,1,1,1,1,...) = A001906

t(A000290) = t(1,4,9,16,25,36,...) = A289779

t(A000027) = t(1,2,3,4,5,6,7,8,...) = A289780

t(A000045) = t(1,2,3,5,8,13,21,...) = A289781

t(A000032) = t(2,1,3,4,7,11,14,...) = A289782

t(A000244) = t(1,3,9,27,81,243,...) = A289783

t(A000302) = t(1,4,16,64,256,...) = A289784

t(A000351) = t(1,5,25,125,625,...) = A289785

t(A005408) = t(1,3,5,7,9,11,13,...) = A289786

t(A005843) = t(2,4,6,8,10,12,14,...) = A289787

t(A016777) = t(1,4,7,10,13,16,...) = A289789

t(A016789) = t(2,5,8,11,14,17,...) = A289790

t(A008585) = t(3,6,9,12,15,18,...) = A289795

t(A000217) = t(1,3,6,10,15,21,...) = A289797

t(A000225) = t(1,3,7,15,31,63,...) = A289798

t(A000578) = t(1,8,27,64,625,...) = A289799

t(A000984) = t(1,2,6,20,70,252,...) = A289800

t(A000292) = t(1,4,10,20,35,56,...) = A289801

t(A002620) = t(1,2,4,6,9,12,16,...) = A289802

t(A001906) = t(1,3,8,21,55,144,...) = A289803

t(A001519) = t(1,1,2,5,13,34,...) = A289804

t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805

t(A008619) = t(1,1,2,2,3,3,4,4,...) = A289806

t(A080513) = t(1,2,2,3,3,4,4,5,...) = A289807

t(A133622) = t(1,2,1,3,1,4,1,5,...) = A289809

t(A000108) = t(1,1,2,5,14,42,...) = A081696

t(A081696) = t(1,1,3,9,29,97,...) = A289810

t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843

t(A175676) = t(1,0,0,2,0,0,3,0,...) = A289844

t(A079977) = t(1,0,1,0,2,0,3,...) = A289845

t(A059841) = t(1,0,1,0,1,0,1,...) = A289846

t(A000040) = t(2,3,5,7,11,13,...) = A289847

t(A008578) = t(1,2,3,5,7,11,13,...) = A289828

t(A000142) = t(1!, 2!, 3!, 4!, ...) = A289924

t(A000201) = t(1,3,4,6,8,9,11,...) = A289925

t(A001950) = t(2,5,7,10,13,15,...) = A289926

t(A014217) = t(1,2,4,6,11,17,29,...) = A289927

t(A000045*) = t(0,1,1,2,3,5,...) = A289975 (* indicates prepended 0's)

t(A000045*) = t(0,0,1,1,2,3,5,...) = A289976

t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977

t(A290990*) = t(0,1,2,3,4,5,...) = A290990

t(A290990*) = t(0,0,1,2,3,4,5,...) = A290991

t(A290990*) = t(0,0,01,2,3,4,5,...) = A290992

Also numerator of e(n-1,n-1) (see Maple line).

Leading coefficient of normalized Legendre polynomial.

Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).

Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005

This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007

Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009

It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - James R. Buddenhagen, Jan 20 2010

The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.

a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011

a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012

a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013

Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016

From Ralf Steiner, Apr 07 2017: (Start)

By analytic continuation to the entire complex plane there exist regularized values for divergent sums:

a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).

Sum_{k>=0} a(k)/A060818(k) = -i.

Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3).

Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3).

a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).

Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2).

Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End)

a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017

Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/A344402. - Peter Luschny, May 20 2021

Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022

a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023

a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023

The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5). - N. J. A. Sloane, Jan 21 2009

T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix. - Michael Somos, Jul 24 2002

p divides a((p-3)/2) for p=11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, ...: A097933. Also primes congruent to {1, 2, 3, 11} mod 12 or primes p such that 3 is a square mod p (excluding 2 and 3) A038874. - Alexander Adamchuk, Jul 05 2006

Partial sums of the even central binomial coefficients. For p prime >=5, a(p-1) = 1 or -1 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). - David Callan, Nov 29 2007

First column of triangle A187887. - Michel Marcus, Jun 23 2013

From Gus Wiseman, Apr 20 2023: (Start)

Also the number of nonempty subsets of {1,...,2n+1} with median n+1, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The odd/even-length cases are A000984 and A006134(n-1). For example, the a(0) = 1 through a(2) = 9 subsets are:

{1} {2} {3}

{1,3} {1,5}

{1,2,3} {2,4}

{1,3,4}

{1,3,5}

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Alternatively, a(n-1) is the number of nonempty subsets of {1,...,2n-1} with median n.

(End)

Also diagonal of rational function R(x,y,z) = 1 /(1 - x - y - z - x*y + x*z).

Annihilating differential operator: x*(16*x^2-24*x+1)*(d/dx)^2 + (48*x^2-48*x+1)*(d/dx) + 12*x-6.

Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - Emeric Deutsch, Nov 23 2003

Product of A007318 and A105868. - Paul Barry, Nov 15 2005

Number of partitions that fit in an n X n box with Durfee square k. - Franklin T. Adams-Watters, Feb 20 2006

From Peter Bala, Oct 23 2008: (Start)

Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).

Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.

(End)

The n-th row consists of the coefficients of the polynomial P_n(t) = Integral_{s = 0..2*Pi} (1 + t^2 - 2*t*cos(s))^n/Pi/2 ds. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - Theodore Kolokolnikov, Oct 26 2010

Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013

From Colin Defant, Sep 16 2018: (Start)

Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.

T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)

The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - Günter Rote, Feb 27 2019

Let P be the poset [n] X [n] ordered by the product order. T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - Geoffrey Critzer, Mar 28 2020

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

Start off with 0 balls in a box. Find the number of ways you can throw 3 balls back out. Then continue to throw 4 balls into the box after each stage. (I.e., the first stage is 0. Then at the next stage there are 4 ways to throw 3 balls back out.) - Ruppi Rana (ruppirana007(AT)hotmail.com), Mar 03 2004

Central coefficients of A094527. - Paul Barry, Mar 08 2011

This is the case m = 2n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012

A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. - Tom Copeland, Oct 10 2012

Conjecture: a(n) == 4 (mod n^3) iff n is prime. - Gary Detlefs, Apr 03 2013

For prime p, the congruence a(p) = binomial(4*p,p) = 4 (mod p^3) is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014