A126120 -id:A126120 - cp's OEIS frontend

A005043 Riordan numbers: a(n) = (n-1)(2a(n-1) + 3*a(n-2))/(n+1).

1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140

Offset: 0

N. J. A. Sloane

Also called Motzkin summands or ring numbers.
The old name was "Motzkin sums", used in certain publications. The sequence has the property that Motzkin(n) = A001006(n) = a(n) + a(n+1), e.g., A001006(4) = 9 = 3 + 6 = a(4) + a(5).
Number of 'Catalan partitions', that is partitions of a set 1,2,3,...,n into parts that are not singletons and whose convex hulls are disjoint when the points are arranged on a circle (so when the parts are all pairs we get Catalan numbers). - Aart Blokhuis (aartb(AT)win.tue.nl), Jul 04 2000
Number of ordered trees with n edges and no vertices of outdegree 1. For n > 1, number of dissections of a convex polygon by nonintersecting diagonals with a total number of n+1 edges. - Emeric Deutsch, Mar 06 2002
Number of Motzkin paths of length n with no horizontal steps at level 0. - Emeric Deutsch, Nov 09 2003
Number of Dyck paths of semilength n with no peaks at odd level. Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1), D=(1,-1). Number of Dyck paths of semilength n with no ascents of length 1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUUDDD. - Emeric Deutsch, Dec 05 2003
Arises in Schubert calculus as follows. Let P = complex projective space of dimension n+1. Take n projective subspaces of codimension 3 in P in general position. Then a(n) is the number of lines of P intersecting all these subspaces. - F. Hirzebruch, Feb 09 2004
Difference between central trinomial coefficient and its predecessor. Example: a(6) = 15 = 141 - 126 and (1 + x + x^2)^6 = ... + 126*x^5 + 141*x^6 + ... (Catalan number A000108(n) is the difference between central binomial coefficient and its predecessor.) - David Callan, Feb 07 2004
a(n) = number of 321-avoiding permutations on [n] in which each left-to-right maximum is a descent (i.e., is followed by a smaller number). For example, a(4) counts 4123, 3142, 2143. - David Callan, Jul 20 2005
The Hankel transform of this sequence give A000012 = [1, 1, 1, 1, 1, 1, 1, ...]; example: Det([1, 0, 1, 1; 0, 1, 1, 3; 1, 1, 3, 6; 1, 3, 6, 15]) = 1. - Philippe Deléham, May 28 2005
The number of projective invariants of degree 2 for n labeled points on the projective line. - Benjamin J. Howard (bhoward(AT)ima.umn.edu), Nov 24 2006
Define a random variable X=trA^2, where A is a 2 X 2 unitary symplectic matrix chosen from USp(2) with Haar measure. The n-th central moment of X is E[(X+1)^n] = a(n). - Andrew V. Sutherland, Dec 02 2007
Let V be the adjoint representation of the complex Lie algebra sl(2). The dimension of the invariant subspace of the n-th tensor power of V is a(n). - Samson Black (sblack1(AT)uoregon.edu), Aug 27 2008
Starting with offset 3 = iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - Gary W. Adamson, Jan 08 2009
a(n) has the following standard-Young-tableaux (SYT) interpretation: binomial(n+1,k)*binomial(n-k-1,k-1)/(n+1)=f^(k,k,1^{n-2k}) where f^lambda equals the number of SYT of shape lambda. - Amitai Regev (amotai.regev(AT)weizmann.ac.il), Mar 02 2010
a(n) is also the sum of the numbers of standard Young tableaux of shapes (k,k,1^{n-2k}) for all 1 <= k <= floor(n/2). - Amitai Regev (amotai.regev(AT)weizmann.ac.il), Mar 10 2010
a(n) is the number of derangements of {1,2,...,n} having genus 0. The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q. Example: a(3)=1 because p=231=(123) is the only derangement of {1,2,3} with genus 0. Indeed, cp'=231*312=123=(1)(2)(3) and so g(p) = (1/2)(3+1-1-3)=0. - Emeric Deutsch, May 29 2010
Apparently: Number of Dyck 2n-paths with all ascents length 2 and no descent length 2. - David Scambler, Apr 17 2012
This is true. Proof: The mapping "insert a peak (UD) after each upstep (U)" is a bijection from all Dyck n-paths to those Dyck (2n)-paths in which each ascent is of length 2. It sends descents of length 1 in the n-path to descents of length 2 in the (2n)-path. But Dyck n-paths with no descents of length 1 are equinumerous with Riordan n-paths (Motzkin n-paths with no flatsteps at ground level) as follows. Given a Dyck n-path with no descents of length 1, split it into consecutive step pairs, then replace UU with U, DD with D, UD with a blue flatstep (F), DU with a red flatstep, and concatenate the new steps to get a colored Motzkin path. Each red F will be (immediately) preceded by a blue F or a D. In the latter case, transfer the red F so that it precedes the matching U of the D. Finally, erase colors to get the required Riordan path. For example, with lowercase f denoting a red flatstep, U^5 D^2 U D^4 U^4 D^3 U D^2 -> (U^2, U^2, UD, DU, D^2, D^2, U^2, U^2 D^2, DU, D^2) -> UUFfDDUUDfD -> UUFFDDUFUDD. - David Callan, Apr 25 2012
From Nolan Wallach, Aug 20 2014: (Start)
Let ch[part1, part2] be the value of the character of the symmetric group on n letters corresponding to the partition part1 of n on the conjucgacy class given by part2. Let A[n] be the set of (n+1) partitions of 2n with parts 1 or 2. Then deleting the first term of the sequence one has a(n) = Sum_{k=1..n+1} binomial(n,k-1)*ch[[n,n], A[n][[k]]])/2^n. This via the Frobenius Character Formula can be interpreted as the dimension of the SL(n,C) invariants in tensor^n (wedge^2 C^n).
Explanation: Let p_j denote sum (x_i)^j the sum in k variables. Then the Frobenius formula says then (p_1)^j_1 (p_2)^j_2 ... (p_r)^j_r is equal to sum(lambda, ch[lambda, 1^j_12^j_2 ... r^j_r] S_lambda) with S_lambda the Schur function corresponding to lambda. This formula implies that the coefficient of S([n,n]) in (((p_1)^1+p_2)/2)^n in its expansion in terms of Schur functions is the right hand side of our formula. If we specialize the number of variables to 2 then S[n,n](x,y)=(xy)^n. Which when restricted to y=x^(-1) is 1. That is it is 1 on SL(2).
On the other hand ((p_1)^2+p_2)/2 is the complete homogeneous symmetric function of degree 2 that is tr(S^2(X)). Thus our formula for a(n) is the same as that of Samson Black above since his V is the same as S^2(C^2) as a representation of SL(2). On the other hand, if we multiply ch(lambda) by sgn you get ch(Transpose(lambda)). So ch([n,n]) becomes ch([2,...,2]) (here there are n 2's). The formula for a(n) is now (1/2^n)*Sum_{j=0..n} ch([2,..,2], 1^(2n-2j) 2^j])*(-1)^j)*binomial(n,j), which calculates the coefficient of S_(2,...,2) in (((p_1)^2-p_2)/2)^n. But ((p_1)^2-p_2)/2 in n variables is the second elementary symmetric function which is the character of wedge^2 C^n and S_(2,...,2) is 1 on SL(n).
(End)
a(n) = number of noncrossing partitions (A000108) of [n] that contain no singletons, also number of nonnesting partitions (A000108) of [n] that contain no singletons. - David Callan, Aug 27 2014
From Tom Copeland, Nov 02 2014: (Start)
Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)].
Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
Various relations among the o.g.f.s may be easily constructed, such as Fib[-Mot(-x)] = -P[P(-x)] = x/(1-2*x) a generating fct for 2^n.
Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1]. (End)
Consistent with David Callan's comment above, A249548, provides a refinement of the Motzkin sums into the individual numbers for the non-crossing partitions he describes. - Tom Copeland, Nov 09 2014
The number of lattice paths from (0,0) to (n,0) that do not cross below the x-axis and use up-step=(1,1) and down-steps=(1,-k) where k is a positive integer. For example, a(4) = 3: [(1,1)(1,1)(1,-1)(1,-1)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,1)(1,-3)]. - Nicholas Ham, Aug 19 2015
A series created using 2*(a(n) + a(n+1)) + (a(n+1) + a(n+2)) has Hankel transform of F(2n), offset 3, F being a Fibonacci number, A001906 (Empirical observation). - Tony Foster III, Jul 30 2016
The series a(n) + A001006(n) has Hankel transform F(2n+1), offset n=1, F being the Fibonacci bisection A001519 (empirical observation). - Tony Foster III, Sep 05 2016
The Rubey and Stump reference proves a refinement of a conjecture of René Marczinzik, which they state as: "The number of 2-Gorenstein algebras which are Nakayama algebras with n simple modules and have an oriented line as associated quiver equals the number of Motzkin paths of length n. Moreover, the number of such algebras having the double centraliser property with respect to a minimal faithful projective-injective module equals the number of Riordan paths, that is, Motzkin paths without level-steps at height zero, of length n." - Eric M. Schmidt, Dec 16 2017
A connection to the Thue-Morse sequence: (-1)^a(n) = (-1)^A010060(n) * (-1)^A010060(n+1) = A106400(n) * A106400(n+1). - Vladimir Reshetnikov, Jul 21 2019
Named by Bernhart (1999) after the American mathematician John Riordan (1903-1988). - Amiram Eldar, Apr 15 2021

			a(5)=6 because the only dissections of a polygon with a total number of 6 edges are: five pentagons with one of the five diagonals and the hexagon with no diagonals.
G.f. = 1 + x^2 + x^3 + 3*x^4 + 6*x^5 + 15*x^6 + 36*x^7 + 91*x^8 + 232*x^9 + ...
From _Gus Wiseman_, Nov 15 2022: (Start)
The a(0) = 1 through a(6) = 15 lone-child-avoiding (no vertices of outdegree 1) ordered rooted trees with n + 1 vertices (ranked by A358376):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((oo)o)  ((oo)oo)  ((oo)ooo)
                     (o(oo))  ((ooo)o)  ((ooo)oo)
                              (o(oo)o)  ((oooo)o)
                              (o(ooo))  (o(oo)oo)
                              (oo(oo))  (o(ooo)o)
                                        (o(oooo))
                                        (oo(oo)o)
                                        (oo(ooo))
                                        (ooo(oo))
                                        (((oo)o)o)
                                        ((o(oo))o)
                                        ((oo)(oo))
                                        (o((oo)o))
                                        (o(o(oo)))
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Katharine Adamyk, Scott Balchin, Miguel Barrero, Steven Scheirer, Noah Wisdom, and Valentina Zapata Castro, On minimal bases in homotopical combinatorics, arXiv:2506.11159 [math.AT], 2025. See p. 16.
Prarit Agarwal and June Nahmgoong, Singlets in the tensor product of an arbitrary number of Adjoint representations of SU(3), arXiv:2001.10826 [math.RT], 2020.
Oswin Aichholzer, Andrei Asinowski, and Tillmann Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
Martin Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
Gert Almkvist, Warren Dicks, and Edward Formanek, Hilbert series of fixed free algebras and noncommutative classical invariant theory, J. Algebra 93 (1985), no. 1, 189-214.
D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. [Annotated scanned copy]
Jean Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 20.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Paul Barry, On the Central Antecedents of Integer (and Other) Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
Paul Barry and Aoife Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
Frank R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
Frank R. Bernhart & N. J. A. Sloane, Correspondence, 1977.
Frank R. Bernhart & N. J. A. Sloane, Emails, April-May 1994.
Jacob L. Bourjaily, Michael Plesser, and Cristian Vergu, The Many Colours of Amplitudes, arXiv:2412.21189 [hep-th], 2024. See pp. 12, 52.
Tom Braden and Artem Vysogorets, Kazhdan-Lusztig polynomials of matroids under deletion, arXiv:1909.09888 [math.CO], 2019.
Naim L. Braha and Toufik Mansour, Some properties of new sequence spaces based on Riordan numbers, Journal of Mathematical Analysis and Applications, Volume 543, Issue 2, Part 1, 128902, (2025).
David Callan, Riordan numbers are differences of trinomial coefficients, September 25, 2006.
David Callan, Pattern avoidance in "flattened" partitions, Discrete Math., 309 (2009), 4187-4191.
David Callan, Bijections for Dyck paths with all peak heights of the same parity, arXiv:1702.06150 [math.CO], 2017.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], 2024. See p. 11.
William Y. C. Chen, Eva Y. P. Deng, and Laura L. M. Yang, Riordan paths and derangements, arXiv:math/0602298 [math.CO], 2006.
Johann Cigler and Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020.
Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015 (see equation (23)).
Benoit Collins, Ion Nechita, and Deping Ye, The absolute positive partial transpose property for random induced states, arXiv preprint arXiv:1108.1935 [math-ph], 2011.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Rodrigo De Castro, Andrés Ramírez, and José L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013. See also Sci. Annals Comp. Sci. (2014) Vol. XXIV, Issue 1, 137-171. See p. 150.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Robert Donaghey and Louis W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Serge Dulucq and Rodica Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8, 1998, 169-191.
László M. Fehér and András P. Juhász, Plücker formulas using equivariant cohomology of coincident root strata, arXiv:2312.06430 [math.AG], 2023. See p. 21.
Mauricio Fernández, On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials, Journal of Elasticity (2019).
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 11.
Yibo Gao, Principal specializations of Schubert polynomials and pattern containment, arXiv:1910.08872 [math.CO], 2019.
Yousra Ghemit and Moussa Ahmia, Two-Catalan numbers: combinatorial interpretation and log-convexity, UPB Sci. Bull., Series A (2024) Vol 86, Iss. 4. See p. 95.
Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
Taras Goy and Mark Shattuck, Determinant identities for the Catalan, Motzkin and Schröder numbers, Art Disc. Appl. Math. (2023).
Phil Hanlon, Counting interval graphs, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.
Benjamin J. Howard, John Millson, Andrew Snowden, and Ravi Vakil, The projective invariants of ordered points on the line, arXiv:math/0505096 [math.AG], 2005.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 423.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Hana Kim and R. P. Stanley, A refined enumeration of hex trees and related polynomials, Preprint 2015, European Journal of Combinatorics, Volume 54, May 2016, pp. 207-219.
Sergey Kitaev, Pavel Salimov, Christopher Severs, and Henning Ulfarsson, Restricted rooted non-separable planar maps, arXiv preprint arXiv:1202.1790 [math.CO], 2012.
Sergei Kitaev, Pavel Salimov, Christopher Severs and Henning Úlfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, 2012. - From _N. J. A. Sloane_, Jan 01 2013
D. E. Knuth, Letter to L. W. Shapiro, R. K. Guy. N. J. A. Sloane, R. P. Stanley, H. Wilf regarding A001006 and A005043.
Nadav Kohen, Uniform Recurrence in the Motzkin Numbers and Related Sequences mod p, arXiv:2403.00149 [math.CO], 2024.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Boyu Li, http://hdl.handle.net/10012/8179, Master's Thesis, Univ. Waterloo, 2013.
Nanna Holmgaard List, Timothé Romain Léo Melin, Martin van Horn, and Trond Saue, Beyond the electric-dipole approximation in simulations of X-ray absorption spectroscopy: Lessons from relativistic theory, arXiv:2001.10738 [physics.chem-ph], 2020.
Ethan Liu, On the Structure and Generators of the Chromatic Algebra, Primes Conference, MIT (2023), p. 18/23.
Ethan Yi-Heng Liu, On the Structure and Generators of the nth-order Chromatic Algebra, arXiv:2401.06095 [math.CO], 2024.
Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238.
Toufik Mansour and Mark Shattuck, Enumeration of Catalan and smooth words according to capacity, Integers (2025) Vol. 25, Art. No. A5. See p. 25.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Zhousheng Mei and Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018.
J. Menashe, Bijections on Riordan objects. [From Tom Copeland, Nov 07 2014]
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.
Scott Morrison, Emily Peters, and Noah Snyder, Categories generated by a trivalent vertex, arXiv preprint arXiv:1501.06869 [math.QA], 2015.
Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Quasipartition and planar quasipartition algebras, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 50. See p. 11.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Marko Riedel et al., Derivation of the Riordan number asymptotics from an inequality of central trinomial coefficients by three Egorychev residues.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.
Emmanuel Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique.
Emmanuel Royer, Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620.
Martin Rubey and Christian Stump, Double deficiencies of Dyck paths via the Billey-Jockusch-Stanley bijection, arXiv:1708.05092 [math.CO], 2017.
Jesus Salas and Alan D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373; arXiv:0711.1738 [math.QA], 2007-2009. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Louis W. Shapiro & N. J. A. Sloane, Correspondence, 1976.
Louis W. Shapiro and Carol J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS 12 (2009) 09.3.2.
Mark Shattuck, On the zeros of some polynomials with combinatorial coefficients, Annales Mathematicae et Informaticae, 42 (2013) pp. 93-101.
H. C. H. Schubert, Allgemeine Anzahlfunctionen für Kegelschnitte, Flächen und Räume zweiten Grades in n Dimensionen, Math. Annalen, June 1894, Volume 45, Issue 2, pp 153-206.
Hua Sun and Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq. 17 (2014) # 14.5.2
Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 03 2021]
Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords.
Eric Weisstein's World of Mathematics, Isotropic Tensor.
Fujine Yano and Hiroaki Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Row sums of triangle A020474, first differences of A082395.
First diagonal of triangular array in A059346.
Binomial transform of A126930. - Philippe Deléham, Nov 26 2009
The Hankel transform of a(n+1) is A128834. The Hankel transform of a(n+2) is floor((2*n+4)/3) = A004523(n+2). - Paul Barry, Mar 08 2011
The Kn11 triangle sums of triangle A175136 lead to A005043(n+2), while the Kn12(n) = A005043(n+4)-2^(n+1), Kn13(n) = A005043(n+6)-(n^2+9*n+56)*2^(n-2) and the Kn4(n) = A005043(2*n+2) = A099251(n+1) triangle sums are related to the sequence given above. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, May 06 2011
Cf. A187306 (self-convolution), A348210 (column 1).
Cf. A000045, A000108, A000957, A001006, A005717, A005773, A007317, A057078, A091867, A104597, A126120, A178514, A249548, A309303.
Bisections: A099251, A099252.

a005043 n = a005043_list !! n
a005043_list = 1 : 0 : zipWith div
   (zipWith (*) [1..] (zipWith (+)
       (map (* 2) $ tail a005043_list) (map (* 3) a005043_list))) [3..]
-- Reinhard Zumkeller, Jan 31 2012

A005043 := proc(n) option remember; if n <= 1 then 1-n else (n-1)*(2*A005043(n-1)+3*A005043(n-2))/(n+1); fi; end;
Order := 20: solve(series((x-x^2)/(1-x+x^2),x)=y,x); # outputs g.f.

a[0]=1; a[1]=0; a[n_]:= a[n] = (n-1)*(2*a[n-1] + 3*a[n-2])/(n+1); Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jun 14 2005 *)
Table[(-3)^(1/2)/6 * (-1)^n*(3*Hypergeometric2F1[1/2,n+1,1,4/3]+ Hypergeometric2F1[1/2,n+2,1,4/3]), {n,0,32}] (* cf. Mark van Hoeij in A001006 *) (* Wouter Meeussen, Jan 23 2010 *)
RecurrenceTable[{a[0]==1,a[1]==0,a[n]==(n-1) (2a[n-1]+3a[n-2])/(n+1)},a,{n,30}] (* Harvey P. Dale, Sep 27 2013 *)
a[ n_]:= SeriesCoefficient[2/(1+x +Sqrt[1-2x-3x^2]), {x, 0, n}]; (* Michael Somos, Aug 21 2014 *)
a[ n_]:= If[n<0, 0, 3^(n+3/2) Hypergeometric2F1[3/2, n+2, 2, 4]/I]; (* Michael Somos, Aug 21 2014 *)
Table[3^(n+3/2) CatalanNumber[n] (4(5+2n)Hypergeometric2F1[3/2, 3/2, 1/2-n, 1/4] -9 Hypergeometric2F1[3/2, 5/2, 1/2 -n, 1/4])/(4^(n+3) (n+1)), {n, 0, 31}] (* Vladimir Reshetnikov, Jul 21 2019 *)
Table[Sqrt[27]/8 (3/4)^n CatalanNumber[n] Hypergeometric2F1[1/2, 3/2, 1/2 - n, 1/4], {n, 0, 31}] (* Jan Mangaldan, Sep 12 2021 *)

a[0]:1$
a[1]:0$
a[n]:=(n-1)*(2*a[n-1]+3*a[n-2])/(n+1)$
makelist(a[n],n,0,12); /* Emanuele Munarini, Mar 02 2011 */

{a(n) = if( n<0, 0, n++; polcoeff( serreverse( (x - x^3) / (1 + x^3) + x * O(x^n)), n))}; /* Michael Somos, May 31 2005 */

my(N=66); Vec(serreverse(x/(1+x*sum(k=1,N,x^k))+O(x^N))) \\ Joerg Arndt, Aug 19 2012

from functools import cache
@cache
def A005043(n: int) -> int:
    if n <= 1: return 1 - n
    return (n - 1) * (2 * A005043(n - 1) + 3 * A005043(n - 2)) // (n + 1)
print([A005043(n) for n in range(32)]) # Peter Luschny, Nov 20 2022

A005043 = lambda n: (-1)^n*jacobi_P(n,1,-n-3/2,-7)/(n+1)
[simplify(A005043(n)) for n in (0..29)]
# Peter Luschny, Sep 23 2014

def ms():
    a, b, c, d, n = 0, 1, 1, -1, 1
    yield 1
    while True:
        yield -b + (-1)^n*d
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)/((n+1)*(n-1))
        c, d = d, (3*(n-1)*c-(2*n-1)*d)/n
A005043 = ms()
print([next(A005043) for  in range(32)]) # _Peter Luschny, May 16 2016

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A000108(k). a(n) = (1/(n+1)) * Sum_{k=0..ceiling(n/2)} binomial(n+1, k)*binomial(n-k-1, k-1), for n > 1. - Len Smiley. [Comment from Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 02 2010: the latter sum should be over the range k=1..floor(n/2).]
G.f.: (1 + x - sqrt(1-2*x-3*x^2))/(2*x*(1+x)).
G.f.: 2/(1+x+sqrt(1-2*x-3*x^2)). - Paul Peart (ppeart(AT)fac.howard.edu), May 27 2000
a(n+1) + (-1)^n = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0). - Bernhart
a(n) = (1/(n+1)) * Sum_{i} (-1)^i*binomial(n+1, i)*binomial(2*n-2*i, n-i). - Bernhart
G.f. A(x) satisfies A = 1/(1+x) + x*A^2.
E.g.f.: exp(x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - Vladeta Jovovic, Apr 28 2003
a(n) = A001006(n-1) - a(n-1).
a(n+1) = Sum_{k=0..n} (-1)^k*A026300(n, k), where A026300 is the Motzkin triangle.
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(k, floor(k/2)). - Paul Barry, Jan 27 2005
a(n) = Sum_{k>=0} A086810(n-k, k). - Philippe Deléham, May 30 2005
a(n+2) = Sum_{k>=0} A064189(n-k, k). - Philippe Deléham, May 31 2005
Moment representation: a(n) = (1/(2*Pi))*Int(x^n*sqrt((1+x)(3-x))/(1+x),x,-1,3). - Paul Barry, Jul 09 2006
Inverse binomial transform of A000108 (Catalan numbers). - Philippe Deléham, Oct 20 2006
a(n) = (2/Pi)* Integral_{x=0..Pi} (4*cos(x)^2-1)^n*sin(x)^2 dx. - Andrew V. Sutherland, Dec 02 2007
G.f.: 1/(1-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). - Paul Barry, Jan 22 2009
G.f.: 1/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). - Paul Barry, May 16 2009
G.f.: 1/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-... (continued fraction). - Paul Barry, Mar 02 2010
a(n) = -(-1)^n * hypergeom([1/2, n+2],[2],4/3) / sqrt(-3). - Mark van Hoeij, Jul 02 2010
a(n) = (-1)^n*hypergeometric([-n,1/2],[2],4). - Peter Luschny, Aug 15 2012
Let A(x) be the g.f., then x*A(x) is the reversion of x/(1 + x^2*Sum_{k>=0} x^k); see A215340 for the correspondence to Dyck paths without length-1 ascents. - Joerg Arndt, Aug 19 2012 and Apr 16 2013
a(n) ~ 3^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 02 2012
G.f.: 2/(1+x+1/G(0)), where G(k) = 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 05 2013
D-finite (an alternative): (n+1)*a(n) = 3*(n-2)*a(n-3) + (5*n-7)*a(n-2) + (n-2)*a(n-1), n >= 3. - Fung Lam, Mar 22 2014
Asymptotics: a(n) = (3^(n+2)/(sqrt(3*n*Pi)*(8*n)))*(1-21/(16*n) + O(1/n^2)) (with contribution by Vaclav Kotesovec). - Fung Lam, Mar 22 2014
a(n) = T(2*n-1,n)/n, where T(n,k) = triangle of A180177. - Vladimir Kruchinin, Sep 23 2014
a(n) = (-1)^n*JacobiP(n,1,-n-3/2,-7)/(n+1). - Peter Luschny, Sep 23 2014
a(n) = Sum_{k=0..n} C(n,k)*(C(k,n-k)-C(k,n-k-1)). - Peter Luschny, Oct 01 2014
Conjecture: a(n) = A002426(n) - A005717(n), n > 0. - Mikhail Kurkov, Feb 24 2019 [The conjecture is true. - Amiram Eldar, May 17 2024]
a(n) = A309303(n) + A309303(n+1). - Vladimir Reshetnikov, Jul 22 2019
From Peter Bala, Feb 11 2022: (Start)
a(n) = A005773(n+1) - 2*A005717(n) for n >= 1.
Conjectures: for n >= 1, n divides a(2*n+1) and 2*n-1 divides a(2*n). (End)

Thanks to Laura L. M. Yang (yanglm(AT)hotmail.com) for a correction, Aug 29 2004

Name changed to Riordan numbers following a suggestion from Ira M. Gessel. - N. J. A. Sloane, Jul 24 2020

A053121 Catalan triangle (with 0's) read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1, 0, 42, 0, 48, 0, 27, 0, 8, 0, 1, 42, 0, 90, 0, 75, 0, 35, 0, 9, 0, 1, 0, 132, 0, 165, 0, 110, 0, 44, 0, 10, 0, 1, 132, 0, 297, 0, 275, 0, 154, 0, 54, 0, 11, 0

Offset: 0

Wolfdieter Lang

Inverse lower triangular matrix of A049310(n,m) (coefficients of Chebyshev's S polynomials).
Walks with a wall: triangle of number of n-step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.
T(n,m) is the number of left factors of Dyck paths of length n ending at height m. Example: T(4,2)=3 because we have UDUU, UUDU, and UUUD, where U=(1,1) and D=(1,-1). (This is basically a different formulation of the previous - walks with a wall - property.) - Emeric Deutsch, Jun 16 2011
"The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]
G.f. for row polynomials p(n,x) := Sum_{m=0..n} (a(n,m)*x^m): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial).
In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310) is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference.
Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231 and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324 and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432 and 4231. - Emeric Deutsch, Oct 12 2006
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array (c(x^2),xc(x^2)), where c(x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Nov 25 2007
A053121^2 = triangle A145973. Convolved with A001405 = triangle A153585. - Gary W. Adamson, Dec 28 2008
By columns without the zeros, n-th row = A000108 convolved with itself n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then n-th row = coefficients of A^(n+1). - Gary W. Adamson, May 13 2009
Triangle read by rows,product of A130595 and A064189 considered as infinite lower triangular arrays; A053121 = A130595*A064189 = B^(-1)*A097609*B where B = A007318. - Philippe Deléham, Dec 07 2009
From Mark Dols, Aug 17 2010: (Start)
As an upper right triangle, rows represent powers of 5-sqrt(24):
  5 - sqrt(24)^1 = 0.101020514...
  5 - sqrt(24)^2 = 0.010205144...
  5 - sqrt(24)^3 = 0.001030928...
(Divided by sqrt(96) these powers give a decimal representation of the columns of A007318, with 1/sqrt(96) being the middle column.) (End)
T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k (1,0)-steps. Example: T(5,3)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, Jun 01 2011
Let S(N,x) denote the N-th Chebyshev S-polynomial in x (see A049310, cf. [W. Lang]). Then x^n = sum_{k=0..n} T(n,k)*S(k,x). - L. Edson Jeffery, Sep 06 2012
This triangle a(n,m) appears also in the (unreduced) formula for the powers rho(N)^n for the algebraic number over the rationals rho(N) = 2*cos(Pi/N) = R(N, 2), the smallest diagonal/side ratio R in the regular N-gon:
  rho(N)^n = sum(a(n,m)*R(N,m+1),m=0..n), n>=0, identical in N >= 1. R(N,j) = S(j-1, x=rho(N)) (Chebyshev S (A049310)). See a comment on this under A039599 (even powers) and A039598 (odd powers). Proof: see the Sep 06 2012 comment by L. Edson Jeffery, which follows from T(n,k) (called here a(n,k)) being the inverse of the Riordan triangle A049310. - Wolfdieter Lang, Sep 21 2013
The so-called A-sequence for this Riordan triangle of the Bell type (c(x^2), x*c(x^2)) (see comments above) is A(x) = 1 + x^2. This proves the recurrence given in the formula section by Henry Bottomley for a(n, m) = a(n-1, m-1) + a(n-1, m+1) for n>=1 and m>=1, with inputs. The Z-sequence for this Riordan triangle is Z(x) = x which proves the recurrence a(n,0) = a(n-1,1), n>=1, a(0,0) = 1. For A- and Z-sequences for Riordan triangles see the W. Lang link under A006232. - Wolfdieter Lang, Sep 22 2013
Rows of the triangle describe decompositions of tensor powers of the standard (2-dimensional) representation of the Lie algebra sl(2) into irreducibles. Thus a(n,m) is the multiplicity of the m-th ((m+1)-dimensional) irreducible representation in the n-th tensor power of the standard one. - Mamuka Jibladze, May 26 2015
The Riordan row polynomials p(n, x) belong to the Boas-Buck class (see a comment and references in A046521), hence they satisfy the Boas-Buck identity: (E_x - n*1)*p(n, x) = (E_x + 1)*Sum_{j=0..n-1} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)*p(n-1-j, x), for n >= 0, where E_x = x*d/dx (Euler operator). For the triangle a(n, m) this entails a recurrence for the sequence of column m, given in the formula section. - Wolfdieter Lang, Aug 11 2017
From Roger Ford, Jan 22 2018: (Start)
For row n, the nonzero values represent the odd components (loops) formed by n+1 nonintersecting arches above and below the x-axis with the following constraints:  The top has floor((n+3)/2) starting arches at position 1 and the next consecutive odd positions. All other starting top arches are in even positions. The bottom arches are a rainbow of arches.  If the component=1 then the arch configuration is a semimeander solution.
Examples: For row 3 {0, 2, 0, 1} there are 3 arch configurations: 2 arch configurations have a component=1; 1 has a component=3.  c=components, U=top arch starting in odd position, u=top arch starting in an even position, d=ending top arch:
.
  top UuUdUddd c=3   top UdUuUddd c=1     top UdUdUudd c=1
       /\                    /\
      //\\                  /  \
     //  \\                / /\ \                    /\
    //    \\              / /  \ \                  /  \
   ///\  /\\\        /\  / / /\ \ \        /\  /\  / /\ \
   \\\ \/ ///        \ \ \ \/ / / /        \ \ \ \/ / / /
    \\\  ///          \ \ \  / / /          \ \ \  / / /
     \\\///            \ \ \/ / /            \ \ \/ / /
      \\//              \ \  / /              \ \  / /
       \/                \ \/ /                \ \/ /
                          \  /                  \  /
                           \/                    \/
For row 4 {2, 0, 3, 0, 1} there are 6 arch configurations: 2 have a component=1; 3 have a component=3: 1 has a component=1. (End)

			Triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6  7  8  9 10 ...
  0:   1
  1:   0   1
  2:   1   0   1
  3:   0   2   0   1
  4:   2   0   3   0   1
  5:   0   5   0   4   0   1
  6:   5   0   9   0   5   0   1
  7:   0  14   0  14   0   6   0  1
  8:  14   0  28   0  20   0   7  0  1
  9:   0  42   0  48   0  27   0  8  0  1
  10: 42   0  90   0  75   0  35  0  9  0  1
  ... (Reformatted by _Wolfdieter Lang_, Sep 20 2013)
E.g., the fourth row corresponds to the polynomial p(3,x)= 2*x + x^3.
From _Paul Barry_, May 29 2009: (Start)
Production matrix is
  0, 1,
  1, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
Boas-Buck recurrence for column k = 2, n = 6: a(6, 2) = (3/4)*(0 + 2*a(4 ,2) + 0 + 6*a(2, 2)) = (3/4)*(2*3 + 6) = 9. - _Wolfdieter Lang_, Aug 11 2017

J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 3.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
J. Cigler, Some q-analogues of Fibonacci, Lucas and Chebyshev polynomials with nice moments, 2013.
J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, 2013.
J. Cigler, Some notes on q-Gould polynomials, 2013.
Emeric Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 and 498).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359, 2014
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986 (see |F_{l,p}| on page 114). - _N. J. A. Sloane_, Jan 29 2011
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Wolfdieter Lang, Chebyshev S-polynomials: ten applications.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415.
MathOverflow, Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?
A. Nkwanta and A. Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, 16 (2013), #13.9.5.
Karim Ritter von Merkl, Computing colored Khovanov homology, arXiv:2505.03916 [math.QA], 2025. See p. 2.
Frank Ruskey and Mark Weston, Spherical Venn Diagrams with Involutory Isometries, Electronic Journal of Combinatorics, 18 (2011), #P191.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014).
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes.
W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.
Index entries for sequences related to Chebyshev polynomials.

Cf. A008315, A049310, A000108, A001405 (row sums), A145973, A153585, A108786, A037952. Another version: A008313. A039598 and A039599 without zeros, and odd and even numbered rows.

Variant without zero-diagonals: A033184 and with rows reversed: A009766.

a053121 n k = a053121_tabl !! n !! k
a053121_row n = a053121_tabl !! n
a053121_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (tail row ++ [0,0])) [1]
-- Reinhard Zumkeller, Feb 24 2012

T:=proc(n,k): if n+k mod 2 = 0 then (k+1)*binomial(n+1,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 12 2006
F:=proc(l,p) if ((l-p) mod 2) = 1 then 0 else (p+1)*l!/( ( (l-p)/2 )! * ( (l+p)/2 +1)! ); fi; end;
r:=n->[seq( F(n,p),p=0..n)]; [seq(r(n),n=0..15)]; # N. J. A. Sloane, Jan 29 2011
A053121 := proc(n,k) option remember; `if`(k>n or k<0,0,`if`(n=k,1,
procname(n-1,k-1)+procname(n-1,k+1))) end proc:
seq(print(seq(A053121(n,k), k=0..n)),n=0..12); # Peter Luschny, May 01 2011

a[n_, m_] /; n < m || OddQ[n-m] = 0; a[n_, m_] = (m+1) Binomial[n+1, (n-m)/2]/(n+1); Flatten[Table[a[n, m], {n, 0, 12}, {m, 0, n}]] [[1 ;; 90]] (* Jean-François Alcover, May 18 2011 *)
T[0, 0] := 1; T[n_, k_]/;0<=k<=n := T[n, k] = T[n-1, k-1]+T[n-1, k+1]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Oliver Seipel, Dec 31 2024 *)

T(n, m)=if(nCharles R Greathouse IV, Mar 09 2016

def A053121_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1] + M[n-1,k+1]
    return M
A053121_triangle(13) # Peter Luschny, Sep 19 2012

a(n, m) := 0 if n

a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n

G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108.

G.f.: G(t,z) = c(z^2)/(1 - t*z*c(z^2)), where c(z) = (1 - sqrt(1-4*z))/(2*z) is the g.f. for the Catalan numbers (A000108). - Emeric Deutsch, Jun 16 2011

a(n, m) = a(n-1, m-1) + a(n-1, m+1) if n > 0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m > 0, a(n, m)=0 if m < 0. - Henry Bottomley, Jan 25 2001

Sum_{k>=0} T(m,k)^2 = A000108(m). - Paul D. Hanna, Apr 23 2005

Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even. - Philippe Deléham, May 26 2005

T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x). - Paul Barry, Feb 16 2006

Sum_{k=0..n} T(n,k)*(k+1) = 2^n. - Philippe Deléham, Mar 22 2007

Sum_{j>=0} T(n,j)*binomial(j,k) = A054336(n,k). - Philippe Deléham, Mar 30 2007

T(2*n+1,2*k+1) = A039598(n,k), T(2*n,2*k) = A039599(n,k). - Philippe Deléham, Apr 16 2007

Sum_{k=0..n} T(n,k)^x = A000027(n+1), A001405(n), A000108(n), A003161(n), A129123(n) for x = 0,1,2,3,4 respectively. - Philippe Deléham, Nov 22 2009

Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Nov 28 2009

Sum_{k=0..n} T(n,k)*A000045(k+1) = A098615(n). - Philippe Deléham, Feb 03 2012

Recurrence for row polynomials C(n, x) := Sum_{m=0..n} a(n, m)*x^m = x*Sum_{k=0..n} Chat(k)*C(n-1-k, x), n >= 0, with C(-1, 1/x) = 1/x and Chat(k) = A000108(k/2) if n is even and 0 otherwise. From the o.g.f. of the row polynomials: G(z; x) := Sum_{n >= 0} C(n, x)*z^n = c(z^2)*(1 + x*z*G(z, x)), with the o.g.f. c of A000108. - Ahmet Zahid KÜÇÜK and Wolfdieter Lang, Aug 23 2015

The Boas-Buck recurrence (see a comment above) for the sequence of column m is: a(n, m) = ((m+1)/(n-m))*Sum_{j=0..n-1-m} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)* a(n-1-j, k), for n > m >= 0 and input a(m, m) = 1. - Wolfdieter Lang, Aug 11 2017

Sum_{m=1..n} a(n,m) = A037952(n). - R. J. Mathar, Sep 23 2021

Edited by N. J. A. Sloane, Jan 29 2011

A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 0, 70, 0, 252, 0, 924, 0, 3432, 0, 12870, 0, 48620, 0, 184756, 0, 705432, 0, 2704156, 0, 10400600, 0, 40116600, 0, 155117520, 0, 601080390, 0, 2333606220, 0, 9075135300, 0, 35345263800, 0, 137846528820, 0, 538257874440, 0, 2104098963720, 0, 8233430727600, 0, 32247603683100, 0, 126410606437752, 0

Offset: 0

Philippe Deléham, Mar 16 2007

Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
From Andrew V. Sutherland, Feb 29 2008: (Start)
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.
Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].
Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.
(End)
From Paul Barry, Aug 10 2009: (Start)
The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).
The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)
a(n) = A204293(2*n,n): central terms of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012
a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012

			a(4) = 6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.

This is A000984 with interspersed zeros. m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

Cf. A107430, A061554, A126120.

a126869 n = a204293_row (2*n) !! n  -- Reinhard Zumkeller, Jan 14 2012

seq((-1)^(n/2)*pochhammer(-n,n/2)/(n/2)!, n=0..43); # Peter Luschny, May 17 2013
seq(n!*coeff(series(hypergeom([],[1],x^2),x,n+1),x,n),n=0..42); # Peter Luschny, Jan 31 2015

Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n,-n},{1},-1],{n,0,30}] (* Peter Luschny, Nov 01 2011 *)

A126869 = lambda n: (2^(n-1)*((-1)^n+1)*gamma((n+1)/2))/(sqrt(pi)*gamma((n+2)/2))
[A126869(n) for n in range(44)] # Peter Luschny, Sep 10 2014

From Andrew V. Sutherland, Feb 29 2008: (Start)
a(2*n) = binomial(2*n,n) = A000984(n); a(2*n+1) = 0.
a(n) = Sum_{k = 0..n} A107430(n,k)*(-1)^(n-k).
a(n) = Sum_{k = 0..n} A061554(n,k)*(-1)^k.
a(n) = (1/Pi)*Integral_{t = 0..Pi} cos^n(t) dt. (End)
E.g.f.: I_0(2*x) where I_n(x) is the modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 10 2011
G.f.: A(x) = 1/sqrt(1 - 4*x^2). - Vladimir Kruchinin, Apr 16 2011
a(n) = (1/Pi)*Integral{x = -2..2} x^n/sqrt((2 - x)*(2 + x)). - Peter Luschny, Sep 12 2011
a(n) = (-1)^floor(n/2) * Hypergeometric([-n,-n],[1], -1). - Peter Luschny, Nov 01 2011
E.g.f.: E(0)/(1 - x) where E(k) = 1 - x/(1 - x/(x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: 1 + x^2/(Q(0) - x^2), where Q(k) = x^2 + (k+1)^2 - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
G.f.: 1/(1 - 2*x^2*Q(0)), where Q(k) = 1 + (4*k+1)*x^2/(k+1 - x^2*(2*k+2)*(4*k+3)/(2*x^2*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: G(0)/(1+x), where G(k) = 1 + x*(2+5*x)*(4*k+1)/((4*k+2)*(1+x)^2 - 2*(2*k+1)*(4*k+3)*x*(2+5*x)*(1+x)^2/((4*k+3)*x*(2+5*x) + 4*(k+1)*(1+x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
a(n) = 2^n*JacobiP(n,0,-1/2-n,-3). - Peter Luschny, Aug 02 2014
a(n) = (2^(n-1)*((-1)^n+1)*Gamma((n+1)/2))/(sqrt(Pi)*Gamma((n+2)/2)). - Peter Luschny, Sep 10 2014
a(n) = n!*[x^n]hypergeom([],[1],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([1/2,-n],[1],2). - Peter Luschny, Feb 03 2015
From Peter Bala, Jul 25 2016: (Start)
a(n) = (-1)^floor(n/2)*Sum_{k = 0..n} (-1)^k*binomial(n,k)^2.
D-finite with recurrence: a(n) = 4*(n - 1)/n * a(n-2) with a(0) = 1, a(1) = 0. (End)
From Ilya Gutkovskiy, Jul 25 2016: (Start)
Inverse binomial transform of A002426.
a(n) = Sum_{k=0..n} (-1)^k*A128014(k).
a(n) ~ 2^n*((-1)^n + 1)/sqrt(2*Pi*n). (End)

A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

Original entry on oeis.org

0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 0

Gerald McGarvey, Jan 20 2005

Called X(n) (i.e., Chi(n)) in Hardy and Wright (p. 241), who show that X(n*m) = X(n)*X(m) for all n and m (i.e., X(n) is completely multiplicative) since (n*m - 1)/2 - (n - 1)/2 - (m - 1)/2 = (n - 1)*(m - 1)/2 == 0 (mod 2) when n and m are odd.
Same as A056594 but with offset 1.
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the non-principal Dirichlet character mod 4. (The principal character is A000035.)
Associated Dirichlet L-functions are for example L(1,chi) = Sum_{n>=1} a(n)/n = A003881, or L(2,chi) = Sum_{n>=1} a(n)/n^2 = A006752, or L(3,chi) = Sum_{n>=1} a(n)/n^3 = A153071. (End)
a(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 0, y = -1, z is arbitrary. - Michael Somos, Nov 27 2019

			G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=4, Chi_2(n).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241.

Muniru A Asiru, Table of n, a(n) for n = 0..1000 (a(0) prepended by Jianing Song)
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A002654, A009116, A009454, A056594, A108520, A111335, A126120, A146559.

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), this sequence (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).

a := [1, 0];; for n in [3..10^2] do a[n] := a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // G. C. Greubel, Aug 23 2018

a := n -> `if`(n mod 2=0, 0, (-1)^((n-1)/2)):
seq(a(n), n=1..10^3); # Muniru A Asiru, Feb 02 2018

a[ n_] := {1, 0, -1, 0}[[ Mod[ n, 4, 1]]]; (* Michael Somos, Jan 13 2014 *)
LinearRecurrence[{0, -1}, {1, 0}, 75] (* G. C. Greubel, Aug 23 2018 *)

{a(n) = if( n%2, (-1)^(n\2))}; /* Michael Somos, Sep 02 2005 */

{a(n) = kronecker( -4, n)}; /* Michael Somos, Mar 30 2012 */

def A101455(n): return (0,1,0,-1)[n&3] # Chai Wah Wu, Jun 21 2024

Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - Mitch Harris May 17 2005
Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos, Sep 02 2005
G.f.: (x - x^3)/(1 - x^4) = x/(1 + x^2). - Michael Somos, Sep 02 2005
G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 * (1 + 2*v). - Michael Somos, Aug 04 2011
a(n + 4) = a(n), a(n + 2) = a(-n) = -a(n), a(2*n) = 0, a(2*n + 1) = (-1)^n for all n in Z. - Michael Somos, Aug 04 2011
a(n + 1) = A056594(n). - Michael Somos, Jan 13 2014
REVERT transform is A126120. STIRLING transform of A009454. BINOMIAL transform is A146559. BINOMIAL transform of A009116. BIN1 transform is A108520. MOBIUS transform of A002654. EULER transform is A111335. - Michael Somos, Mar 30 2012
Completely multiplicative with a(p) = 2 - (p mod 4). - Werner Schulte, Feb 01 2018
a(n) = (-(n mod 2))^binomial(n, 2). - Peter Luschny, Sep 08 2018
a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - Jianing Song, Sep 09 2018
From Jianing Song, Nov 14 2018: (Start)
a(n) = ((-4)/n) (or more generally, ((-4^i)/n) for i > 0), where (k/n) is the Kronecker symbol.
E.g.f.: sin(x).
Dirichlet g.f. is the Dirichlet beta function.
a(n) = A091337(n)*A188510(n). (End)

a(0) prepended by Jianing Song, Nov 14 2024

A010790 a(n) = n!*(n+1)!.

Original entry on oeis.org

1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000

Offset: 0

N. J. A. Sloane

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002
If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009
a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011
a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011
a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n-1) is the number of elements of the wreath product of S_n and S_2 with cycle partition equal to (2n), where S_n is the symmetric group of order n. - Josaphat Baolahy, Mar 12 2024

			G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.
Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [Ross La Haye, Jan 06 2009]

T. D. Noe, Table of n, a(n) for n = 0..100
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
Steve Gadbois, 104.12 From calendar coincidence to factorials to Ramanujan, The Mathematical Gazette (2020) Vol. 104, Issue 560, 304-306.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 268.
S. Tanimoto, Parity alternating permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).
Index entries for sequences related to factorial numbers

Second column of triangle A129065.

Cf. A004737, A000290, A000108, A126120.

[Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
f:= n-> n!*(n+1)!: seq(f(n), n=0..30);
```

s=1;lst={s};Do[s+=(s*=n)*n;AppendTo[lst, s], {n, 1, 4!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Times@@@Partition[Range[0,25]!,2,1] (* Harvey P. Dale, Jun 17 2011 *)

a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011

from math import factorial
def A010790(n): return factorial(n)**2*(n+1) # Chai Wah Wu, Apr 22 2024

[stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009

From Karol A. Penson, Oct 23 2001: (Start)
Integral representation as n-th moment of a positive function f on the positive half axis, where f(x) = 2*sqrt(x)*BesselK(1, 2*sqrt(x)). Then:
a(n) = Integral_{x>=0} x^n * f(x) dx.
G.f.: a(0) = 1 and a(n) = subs(x=0, n!*diff(1/((x-1)^2), x$n)) for n >= 1. (End)
Sum_{i >=0} 1/a(i) = A096789. - Gerald McGarvey, Jun 10 2004
With b(n)=A002378(n) for n>0 and b(0)=1, a(n) = b(n)*b(n-1)...*b(0). - Tom Copeland, Sep 21 2011
a(n) = det(PS(i+1,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
a(n) = (2*n)! / A000108(n) which implies that the e.g.f. of A126120 is Sum_{k>=0} x^(2*k) / a(k). - Michael Somos, Nov 15 2014
0 = a(n)*(+18*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Nov 15 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... = A348607 (End)
D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - R. J. Mathar, Jan 27 2020
a(n) = 1/([x^n] hypergeom([], [2], x)). - Peter Luschny, Sep 13 2024

A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

Original entry on oeis.org

1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500

Offset: 0

N. J. A. Sloane

nonn

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003
Number of tilings of an  hexagon.
a(n) is the number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan, Jul 25 2005
The number of returning walks of length 2n on the upper half of a square lattice, since a(n) = Sum_{k=0..2n} binomial(2n,k)*A126120(k)*A126869(n-k). - Andrew V. Sutherland, Mar 24 2008
For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). - Peter Bala, Oct 14 2008
The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012
a(n) is number of pairs of non-intersecting lattice paths from (0,0) to (n+1,n+1) using (1,0) and (0,1) as steps. Here, non-intersecting means two paths do not share a vertex except the origin and the destination. For example, a(1) = 3 because we have three such pairs from (0,0) to (2,2): {NNEE,EENN}, {NNEE,ENEN}, {NENE,EENN}. - Ran Pan, Oct 01 2015
Also the number of ordered rooted trees with 2(n+1) nodes and n+1 leaves, i.e., half of the nodes are leaves. These trees are ranked by A358579. The unordered version is A185650. - Gus Wiseman, Nov 27 2022
The number of secondary GL(2) invariants constructed from n+1 two component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025

			G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
From _Gus Wiseman_, Nov 27 2022: (Start)
The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
  (((o)oo))  (((o)o)o)  (((o))oo)
  (((oo)o))  (((oo))o)  ((o)(o)o)
  (((ooo)))  ((o)(oo))  ((o)o(o))
  ((o(o)o))  ((o(o))o)  (o((o))o)
  ((o(oo)))  ((oo)(o))  (o(o)(o))
  ((oo(o)))  (o((o)o))  (oo((o)))
             (o((oo)))
             (o(o(o)))
(End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Elena Barcucci, Andrea Frosini and Simone Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838.
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Ivan Marin and Emmanuel Wagner, A cubic defining algebra for the Links-Gould polynomial. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From _N. J. A. Sloane_, Sep 21 2012
Henri Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Guoce Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A000356, A010370, A038535.
Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008
Cf. A000142, A000894, A248045.
Equals half of A267981.
Counts the trees ranked by A358579.
A001263 counts ordered rooted trees by nodes and leaves.
A090181 counts ordered rooted trees by nodes and internals.
Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.

a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013

[Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

with(combstruct): bin := {B=Union(Z,Prod(B,B))} :seq(1/2*binomial(2*i,i)*(count([B,bin,unlabeled],size=i)), i=1..18) ; # Zerinvary Lajos, Jun 06 2007

a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ BesselI[0, 2 x] BesselI[1, 2 x] / x, {x, 0, 2 n}]]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* Michael Somos, Sep 18 2016 *)
a[n_] := (2 n + 1) CatalanNumber[n]^2;
Array[a, 20, 0] (* Peter Luschny, Mar 03 2020 *)

{a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */

a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ Hugo Pfoertner, Oct 22 2022

-4*a(n) = A010370(n+1).
G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. [edited by Olivier Gérard, Feb 16 2011]
G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x) * BesselI(1, 2*x) / x. - Michael Somos, Jun 22 2005
a(n) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007
For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - Gary W. Adamson, Apr 08 2011
a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle).
G.f.: If G_N(x) = 1 + Sum_{k=1..N} ((2*k)!*(2*k+1)!*x^k)/(k!*(k+1)!)^2, G_N(x) = 1 + 12*x/(G(0) - 12*x); G(k) = 16*x*k^2 + 32*x*k + k^2 + 4*k + 12*x + 4 - 4*x*(2*k+3)*(2*k+5)*(k+2)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
D-finite with recurrence (n+1)^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014
a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: 2F2(1/2,3/2; 2,2; 16*x).
a(n) ~ 2^(4*n+1)/(Pi*n^2). (End)
a(n) = A005408(n)*(A000108(n))^2. - Ivan N. Ianakiev, Nov 13 2019
a(n) = det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(n+j+1,i+1). - Benoit Cloitre, Oct 22 2022
a(n) = Integral_{x=0..16} x^n*W(x) dx, where W(x) = (16*EllipticE(1 - x/16) - x*EllipticK(1 - x/16))/(8*Pi^2*sqrt(x)), n=>0. W(x) diverges at x=0, monotonically decreases for x>0, and vanishes at x=16. EllipticE and EllipticK are elliptic functions. This integral representation as n-th moment of a positive function W(x) on the interval [0, 16] is unique. - Karol A. Penson, Dec 20 2023

More terms from Andrew V. Sutherland, Mar 24 2008

A138364 The number of Motzkin n-paths with exactly one flat step.

Original entry on oeis.org

0, 1, 0, 3, 0, 10, 0, 35, 0, 126, 0, 462, 0, 1716, 0, 6435, 0, 24310, 0, 92378, 0, 352716, 0, 1352078, 0, 5200300, 0, 20058300, 0, 77558760, 0, 300540195, 0, 1166803110, 0, 4537567650, 0, 17672631900, 0, 68923264410, 0, 269128937220, 0

Offset: 0

Andrew V. Sutherland, Mar 16 2008

An aerated version of A001700, which is the main entry for this sequence.

Number of paths in the half-plane x>=0, from (0,0) to (n,1), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=3, we have the 3 paths: UUD, UDU, DUU. - José Luis Ramírez Ramírez, Apr 19 2015

			a(5)=10 since the coefficient of z^5 in I_1(2z) is binomial(5,3)=10.

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1999.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic Curves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)

Cf. A001700, A057977, A126869.

&cat[[0, Binomial(n, (n+1) div 2)]: n in [1..50 by 2]]; // Vincenzo Librandi, Apr 20 2015

a[ n_] := SeriesCoefficient[ n! BesselI[ 1, 2 x], {x, 0, n}]; (* Michael Somos, Mar 19 2014 *)

x='x+O('x^66); concat([0], Vec( -(sqrt(1-4*x^2)+2*x^2-1) / (x*sqrt(1-4*x^2)+4*x^3-x))) \\ Joerg Arndt, May 08 2013

from math import comb
def A138364(n): return comb(n,n>>1) if n&1 else 0 # Chai Wah Wu, Aug 25 2025

def A138364(n):
    if is_even(n): return 0
    return binomial(n,n//2)
[A138364(n) for n in (0..42)]  # Peter Luschny, Mar 18 2014

a(n) = binomial(n,(n+1)/2) for n odd, 0 otherwise.
E.g.f.: I_1(2z), where I_1 is the hyperbolic Bessel function of order 1.
a(n) = (1/(2*Pi))*integral(x=-2..2, x^n*x/sqrt((2+x)*(2-x))). - Peter Luschny, Sep 12 2011
G.f.: -(sqrt(1-4*x^2)+2*x^2-1)/(x*sqrt(1-4*x^2)+4*x^3-x). - Vladimir Kruchinin, Mar 08 2013
a(n) + A126120(n) = A057977(n). - Peter Luschny, Mar 18 2014
G.f.: z*C(z^2)/(1-2*z^2*C(z^2)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = Integral_[-Pi,Pi] cos^(n+1)/(2^(n-1)*Pi). - M. F. Hasler, Jul 12 2018

New name is a comment by David Scambler, May 02 2013. - Peter Luschny, Mar 18 2014

A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).

Original entry on oeis.org

1, 1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650

Offset: 0

Labos Elemer, Nov 13 2000

The numbers can be seen as a generalization of the Catalan numbers, extending A000984(n)/(n+1) to A056040(n)/(floor(n/2)+1). They can also be seen as composing the aerated Catalan numbers A126120 with the aerated complementary Catalan numbers A138364. (Thus the name 'extended Catalan numbers' might be apt for this sequence.) - Peter Luschny, May 03 2011
a(n) is the number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and maximally one step H=(1,0). - Alois P. Heinz, Apr 17 2013
Equal to A063549 (see comments in that sequence). - Nathaniel Johnston, Nov 17 2014
a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - Peter Luschny, Feb 23 2019

			This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers.

Cf. A001405, A063549.

Bisections are A000108 and A001700.

A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
(2 + b(z))/sqrt(1-4*z^2) - b(z) end:
seq(coeff(series(A057977_ogf(z),z,n+3),z,n), n = 0..35);
A057977_rec := n -> `if`(n=0, 1, A057977_rec(n-1)*n^modp(n,2)
*(4/(n+2))^modp(n+1,2));
A057977_int := proc(n) int((x^(2*n-1)*((4-x)^2/x)^cos(Pi*n))^(1/4),x=0..4)/(2*Pi); round(evalf(%)) end:
A057977 := n -> (n!/iquo(n,2)!^2) / (iquo(n,2)+1):
seq(A057977(n), n=0..35); # Peter Luschny, Apr 30 2011
b := proc(p, q) option remember; local S;
   if p = 0 and q = 0 then return 1 fi;
   if p < 0 or  p > q then return 0 fi;
   S := b(p-2, q) + b(p, q-2);
   if type(q, odd) then S := S + b(p-1, q-1) fi;
   S end:
seq(b(n, n), n=0..35); # Peter Luschny, Feb 23 2019

a[n_] := n! / (Quotient[n, 2]!^2 * (Quotient[n, 2]+1)); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 03 2012, after Peter Luschny *)

a(n)=if(n<0,0,(n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
a(n)=n!/(n\2)!^2/(n\2+1) \\ Charles R Greathouse IV, May 02 2011

def A057977():
    x, n = 1, 1
    while True:
        yield x
        m = n if is_odd(n) else 4/(n+2)
        x *= m
        n += 1
a = A057977(); [next(a) for i in range(36)]   # Peter Luschny, Oct 21 2013

G.f.: (4*x^2+x-1+(1-x)*sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*x^2). E.g.f.: (1+1/x)*BesselI(1, 2*x). - Vladeta Jovovic, Jan 19 2004
From Peter Luschny, Apr 30 2011: (Start)
Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
Asymptotic formula: Let [n even] = 1 if n is even, 0 otherwise. Let N := n+1+[n even]. Then a(n) ~ 2^N /((n+1)^[n even]*sqrt(Pi*(2*N+1))).
Integral representation: a(n) = (1/(2*Pi))*Int_{x=0..4}(x^(2*n-1)* ((4-x)^2/x)^cos(Pi*n))^(1/4) (End)
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+2)//U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
From R. J. Mathar, Sep 16 2016: (Start)
D-finite with recurrence: (n+2)*a(n) - n*a(n-1) + 4*(-2*n+1)*a(n-2) + 4*(n-1)*a(n-3) + 16*(n-3)*a(n-4) = 0.
D-finite with recurrence: -(n+2)*(n^2-5)*a(n) + 4*(-2*n-1)*a(n-1) + 4*(n-1)*(n^2+2*n-4)*a(n-2) = 0. (End)
Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

A037952 a(n) = binomial(n, floor((n-1)/2)).

Original entry on oeis.org

0, 1, 1, 3, 4, 10, 15, 35, 56, 126, 210, 462, 792, 1716, 3003, 6435, 11440, 24310, 43758, 92378, 167960, 352716, 646646, 1352078, 2496144, 5200300, 9657700, 20058300, 37442160, 77558760, 145422675, 300540195, 565722720, 1166803110, 2203961430, 4537567650

Offset: 0

N. J. A. Sloane

nonn

First differences of central binomial coefficients: a(n) = A001405(n+1) - A001405(n).
The maximum size of an intersecting (or proper) antichain on an n-set. - Vladeta Jovovic, Dec 27 2000
Number of ordered trees with n+1 edges, having root of degree at least 2 and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch, Aug 02 2002
a(n)=number of Dyck (n+1)-paths that are symmetric but not prime. A prime Dyck path is one that returns to the x-axis only at its terminal point. For example a(3)=3 counts UDUUDDUD, UUDDUUDD, UDUDUDUD. - David Callan, Dec 09 2004
Number of involutions of [n+2] containing the pattern 132 exactly once. For example, a(3)=3 because we have 1'3'2'45, 42'5'13' and 52'4'3'1 (the entries corresponding to the pattern 132 are "primed"). - Emeric Deutsch, Nov 17 2005
Also number of ways to put n eggs in floor(n/2) baskets where order of the baskets matters and all baskets have at least 1 egg. - Ben Paul Thurston, Sep 30 2006
For n >= 1 the number of standard Young tableaux with shapes corresponding to partitions into at most 2 distinct parts. - Joerg Arndt, Oct 25 2012
It seems that 3, 4, 10, ... are Colbourn's Covering Array Numbers CAN(2,k,2). - Ryan Dougherty, May 27 2015
Essentially the same as A007007. - Georg Fischer, Oct 02 2018
a(n) is the number of subsets of {1,2,...,n} that contain exactly 1 more odd than even elements.  For example, for n = 6, a(6) = 15 and the 15 sets are {1}, {3}, {5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {3,4,5}, {3,5,6}, {1,2,3,4,5}, {1,2,3,5,6}, {1,3,4,5,6}. - Enrique Navarrete, Dec 21 2019
a(n) is the number of lattice paths of n steps taken from the step set {U=(1,1), D=(1,-1)} that start at the origin, never go below the x-axis, and end strictly above the x-axis; more succinctly, proper left factors of Dyck paths. For example, a(3)=3 counts UUU, UUD, UDU, and a(4)=4 counts UUUU, UUUD, UUDU, UDUU. - David Callan and Emeric Deutsch, Jan 25 2021
For n >= 3, a(n) is also the number of pinnacle sets in the (n-2)-Plummer-Toft graph. - Eric W. Weisstein, Sep 11 2024

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 25.
C. J. Colbourn, Table of CAN(2, k, 2)
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 89-94.
O. Guibert and T. Mansour, Restricted 132-involutions, Sem. Lotharingien de Combinatoire, 48, 2002, Article B48a (Corollary 4.2).
M. Miyakawa, A. Nozaki, G. Pogosyan, and I. G. Rosenberg, A map from the lower-half of the n-Cube onto the (n-1)-Cube which preserves intersecting antichains, Discr. Appl. Math. 92 (2-3) (1999) 223-228.
M. van de Vel, Determination of msd(L^n), J. Algebraic Combin., 9 (1999), 161-171.
Eric Weisstein's World of Mathematics, Pinnacle Set.
Eric Weisstein's World of Mathematics, Plummer-Toft Graph.

Cf. A007007, A032263, A014495 (partial sums), A001405 (partial sums + 1).
Cf. A035951, A035953, A035954, A035955, A035956, A035957.
Cf. A051303, A051304, A051305, A051306, A051307.
Cf. A047171, A036256, A051920.
Cf. A265848.

a037952 n = a037952_list !! n
a037952_list = zipWith (-) (tail a001405_list) a001405_list
-- Reinhard Zumkeller, Mar 04 2012

[Binomial(n, Floor((n-1)/2)): n in [0..40]]; // G. C. Greubel, Jun 21 2022

a:= n-> binomial(n, floor((n-1)/2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 19 2017

Table[ Binomial[n, Floor[n/2]], {n, 0, 35}]//Differences (* Jean-François Alcover, Jun 10 2013 *)
f[n_] := Binomial[n, Floor[(n-1)/2]]; Array[f, 35, 0] (* Robert G. Wilson v, Nov 13 2014 *)

a(n) = binomial(n, (n-1)\2); \\ Altug Alkan, Oct 03 2018

[binomial(n, (n-1)//2) for n in (0..40)] # G. C. Greubel, Jun 21 2022

E.g.f.: BesselI(1, 2*x) + BesselI(2, 2*x). - Vladeta Jovovic, Apr 28 2003
O.g.f.: (1-sqrt(1-4x^2))/(x - 2x^2 + x*sqrt(1-4x^2)).
Convolution of A001405 and A126120 shifted right: g001405(x)*g126120(x) = g037952(x)/x. - Philippe Deléham, Mar 17 2007
D-finite with recurrence: (n+2)*a(n) + (-n-2)*a(n-1) + 2*(-2*n+1)*a(n-2) + 4*(n-2)*a(n-3) = 0. - R. J. Mathar, Jan 25 2013. Proved by Robert Israel, Nov 13 2014
For n > 0: a(n) = A265848(n,0). - Reinhard Zumkeller, Dec 24 2015
a(n) = binomial(n, (n-2)/2) = A001791(n/2), n even; a(n) = binomial(n, (n+1)/2) = A001700((n-1)/2), n odd. - Enrique Navarrete, Dec 21 2019
From R. J. Mathar, Sep 23 2021: (Start)
A001405(n) = a(n) + A000108(n/2), where A(.)=0 for non-integer arguments.
a(n) = Sum_{m=1..n} A053121(n,m) [comment Callan-Deutsch].
a(2n+1) = A000984(n+1)/2. (End)
a(n) = Sum_{k=2..n} A143359(n,k). [Callan's 2004 comment]. - R. J. Mathar, Sep 24 2021
From Amiram Eldar, Sep 27 2024: (Start)
Sum_{n>=1} 1/a(n) = 1 + Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = (3 - Pi/sqrt(3))/9. (End)

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A161006 Convolution of A000108 (Catalan numbers) with A126120 (Catalan numbers interpolated with 0's).

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A005043 Riordan numbers: a(n) = (n-1)*(2*a(n-1) + 3*a(n-2))/(n+1).

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A053121 Catalan triangle (with 0's) read by rows.

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A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

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A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

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A005043 Riordan numbers: a(n) = (n-1)(2a(n-1) + 3*a(n-2))/(n+1).

A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.