A051292 -id:A051292 - cp's OEIS frontend

A004148 Generalized Catalan numbers: a(n+1) = a(n) + Sum_{k=1..n-1} a(k)*a(n-1-k).

1, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374

Offset: 0

N. J. A. Sloane

Arises in enumerating secondary structures of RNA molecules. The 17 structures with 6 nucleotides are shown in the illustration (after Waterman, 1978).
Hankel transform is period 8 sequence [1, 0, -1, -1, -1, 0, 1, 1, ...] (A046980).
Enumerates peak-less Motzkin paths of length n. Example: a(5)=8 because we have HHHHH, HHUHD, HUHDH, HUHHD, UHDHH, UHHDH, UHHHD, UUHDD, where U=(1,1), D=(1,-1) and H=(1,0). - Emeric Deutsch, Nov 19 2003
Number of Dyck paths of semilength n-1 with no UUU's and no DDD's, where U=(1,1) and D=(1,-1) (n>0). - Emeric Deutsch, Nov 19 2003
For n >= 1, a(n) = number of dissections of an (n+2)-gon with strictly disjoint diagonals and no diagonal incident with the base. (One side of the (n+2)-gon is designated the base.) - David Callan, Mar 23 2004
For n >= 2, a(n-2)= number of UU-free Motzkin n-paths = number of DU-free Motzkin n-paths. - David Callan, Jul 15 2004
a(n) = number of UU-free Motzkin n-paths containing no low peaks (A low peak is a UD pair at ground level, i.e., whose removal would create a pair of Motzkin paths). For n >= 1, a(n) = number of UU-free Motzkin (n-1)-paths = number of DU-free Motzkin (n-1)-paths. a(n) is asymptotically ~ c n^(-3/2) (1 + phi)^n with c = 1.1043... and phi=(1+sqrt(5))/2. - David Callan, Jul 15 2004. In closed form, c = sqrt(30+14*sqrt(5))/(4*sqrt(Pi)) = 1.104365547309692849... - Vaclav Kotesovec, Sep 11 2013
a(n) = number of Dyck (n+1)-paths with all pyramid sizes >= 2. A pyramid is a maximal subpath of the form k upsteps immediately followed by k downsteps and its size is k. - David Callan, Oct 24 2004
a(n) = number of Dyck paths of semilength n+1 with no small pyramids (n >= 1). A pyramid is a maximal sequence of the form k Us followed by k Ds with k >= 1. A small pyramid is one with k=1. For example, a(4)=4 counts the following Dyck 5-paths (pyramids denoted by lowercase letters and separated by a vertical bar): uuuuuddddd, Uuudd|uuddD, uudd|uuuddd, uuuddd|uudd. - David Callan, Oct 25 2004
From Emeric Deutsch, Jan 08 2006: (Start)
a(n) = number of Motzkin paths of length n-1 with no peaks at level >= 1. Example: a(4)=4 because we have HHH, HUD, UDH and UHD, where U=(1,1), D=(1,-1) and H=(1,0).
a(n) = number of Motzkin paths of length n+1 with no level steps on the x-axis and no peaks at level >=1. Example: a(4)=4 because we have UHHHD, UHDUD, UDUHD and UUHDD, where U=(1,1), D=(1,-1) and H=(1,0).
a(n) = number of Dyck paths of length 2n having no ascents and descents of even length. An ascent (descent) is a maximal sequence of up (down) steps. Example: a(4)=4 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD and UUUDUDDD, where U=(1,1), D=(1,-1) and H=(1,0).
a(n) = number of Dyck paths of length 2n having ascents only of length 1 or 2 and having no peaks of the form UUDD. An ascent is a maximal sequence of up steps. Example: a(4)=4 because we have UDUDUDUD, UDUUDUDD, UUDUDDUD and UUDUDUDD, where U=(1,1), D=(1,-1) and H=(1,0).
a(n) = number of noncrossing partitions of [n+1] having no singletons and in each block the two leftmost points are of the form i,i+1. Example: a(4)=4 because we have 12345, 12/345, 123/45 and 125/34; the noncrossing partition 145/23 does not satisfy the requirements because 1 and 4 are not consecutive.
a(n) = number of noncrossing partitions of [n+1] with no singletons, except possibly the block /1/ and no blocks of the form /i,i+1/, except possibly the block /1,2/. Example: a(4)=4 because we have 12345, 1/2345, 12/345 and 15/234.
(End)
a(n+1) = [1, 1, 2, 4, 8, 17, 37, ...] gives the antidiagonal sums of triangle of Narayana, A001263. - Philippe Deléham, Oct 21 2006
a(n) = number of Dyck (n+1)-paths with no UDUs and no DUDs. For example, a(4)=4 counts UUUUUDDDDD, UUUDDUUDDD, UUDDUUUDDD, UUUDDDUUDD. - David Callan, May 08 2007
a(n) is also the number of Dyck paths of semilength n without height of peaks and valleys 2(mod 3). - Majun (majun(AT)math.sinica.edu.tw), Nov 29 2008
G.f. of a(n+1) is 1/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-... (continued fraction). - Paul Barry, May 20 2009
A Chebyshev transform of the Motzkin numbers A001006: g.f. is the image of (1-x-(1-2x-3x^2)^(1/2))/(2x^2) under the mapping that takes g(x) to (1/(1+x^2))g(x/(1+x^2)). - Paul Barry, Mar 10 2010
For n >= 1, the number of lattice paths of weight n -1 that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and  a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. a(4)=4 because, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), we have the following four paths of weight 3: hH, Hh, hhh, and ud. (See the g.f. C(x) on p. 295 of the Bona-Knopfmacher reference.)
From David Callan, Aug 27 2014: (Start)
a(n) = number of noncrossing partitions of [n] with all blocks of size 1 or 2 and no blocks of the form /i,i+1/. Example: a(4)=4 because we have 1234, 13/2/4, 14/2/3, and 1/24/3.
It appears that a(n) = number of permutations of [n] that avoid the three dashed patterns 123, 132, 24-13, and contain no small jumps (jumps of one unit). For example, a(4)=4 counts 3214, 3241, 4213, and 4321 but not 4312 because 12 is a small jump. (End)
Number of DU_{k}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - Sergey Kirgizov, Apr 08 2018
a(n) is also the number of 3412-avoiding involutions on [n] with no transpositions of the form (i,i+1). For example, a(4)=4 counts the involutions 1234, 1432, 3214, 4231. - Juan B. Gil, May 23 2020
For n >= 2, a(n) equals the number of Dyck paths with air pockets of length n. A Dyck path with air pockets is a nonempty lattice path in the first quadrant of Z^2 starting at the origin, ending on the x-axis, and consisting of up-steps U = (1,1) and down-steps D_k = (1, -k), k >= 1, where two down-steps cannot be consecutive. For example, the only path of length 2 is UD_1; for length 3 we have UU_D2; for length 4 there are 2 paths: UUUD_3, UD_1UD_1; and for length 5 we have 4 paths: UUUUD_4, UUD_2UD_1, UD_1UUD_2, UUD_1UD_2. - Sergey Kirgizov, Dec 15 2022

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 37*x^7 + 82*x^8 + 185*x^9 + 432*x^10 + ...
Det([1]) = 1, Det([1, 1; 1, 1]) = 0, Det([1, 1, 1; 1, 1, 2; 1, 2, 4]) = -1. - _Michael Somos_, May 12 2022

A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..2404 (first 201 terms from T. D. Noe)
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Sergey Avgustinovich, Sergey Kitaev and Alexander Valyuzhenich, Avoidance of boxed mesh patterns on permutations.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See pp. 22, 27.
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, Conjectures on Somos 4, 6 and 8 sequences using Riordan arrays and the Catalan numbers, arXiv:2211.12637 [math.CO], 2022.
Paul Barry, Aoife Hennessy and Nikolaos Pantelidis, Algebraic properties of Riordan subgroups, J Algebr Comb 53, 1015-1036 (2021).
Antonio Bernini, Matteo Cervetti, Luca Ferrari and Einar Steingrimsson, Enumerative combinatorics of intervals in the Dyck pattern poset, arXiv:1910.00299 [math.CO], 2019.
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
A. J. Bu and Robert Dougherty-Bliss, Enumerating Restricted Dyck Paths with Context-Free Grammars, arXiv:2009.09061 [math.CO], 2020.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Naiomi Cameron and Everett Sullivan, Peakless Motzkin paths with marked level steps at fixed height, Discrete Mathematics 344.1 (2021): 112154.
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
Emeric Deutsch and Sergi Elizalde, Restricted simsun permutations, Ann. Comb. 16, No. 2, 253-269 (2012).
Emeric Deutsch and L. W. Shapiro, A bijection between ordered trees and 2-Motzkin paths and its many consequences, Disc. Math. 256 (2002) 655-670.
R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020.
T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 96.
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2.
Eric S. Egge and Kailee Rubin, Snow Leopard Permutations and Their Even and Odd Threads, arXiv:1508.05310 [math.CO], 2015.
S. B. Ekhad and M. Yang, Automated proofs of many conjectured recurrences in the OEIS made by R. J. Mathar, arXiv:1707.04654 (2017).
S. J. Evans, A. P. Veselov, and B. Winn, Quantum Kronecker fractions, arXiv:2410.15666 [math.NT], 2024. See p. 7.
Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.
Guo-Niu Han and Emmanuel Pedon, Hankel continued fractions and Hankel determinants for q-deformed metallic numbers, arXiv:2502.05993 [math.NT], 2025. See p. 5.
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Eq. 27.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 421
Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, Bulletin of Mathematical Biology 70 (2008), 951-970. See Eq. 22.
Shu-Chung Liu, Jun Ma and Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289. [From Majun (majun(AT)math.sinica.edu.tw), Nov 29 2008]
Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
Sophie Morier-Genoud and Valentin Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes, arXiv:2011.10809 [math.QA], 2020. See section 3.3. Mentions this sequence.
Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, arXiv:2503.23834 [math.CO], 2025. See p. 13.
Emanuele Munarini and Norma Zagaglia Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
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.
Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012
Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 8.
N. J. A. Sloane, Illustration of a(6) = 17 (after Waterman, 1978).
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.
M. S. Waterman, Home Page (contains copies of his papers)
M. S. Waterman, Secondary structure of single-stranded nucleic acids, Studies in Foundations and Combinatorics, Vol. 1, pp. 167-212, 1978.

Second row of A064645.
Cf. A000108, A001006, A088518, A051292 (lgf), A004149, A131198.
Cf. A046980 (Hankel transform).

a004148 n = a004148_list !! n
a004148_list = 1 : f [1] where
f xs'@(x:xs) = y : f (y : xs') where
y = x + sum (zipWith (*) xs $ reverse $ tail xs)
-- Reinhard Zumkeller, Nov 13 2012

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( (1-x+x^2 - Sqrt(1-2*x-x^2-2*x^3+x^4))/(2*x^2) )); // G. C. Greubel, Dec 30 2019

w := proc(l) x - 1 - x^2*(1 - x^l)/(1-x) end:
S := proc(l) (-w(l) - sqrt(w(l)^2 - 4*x^2))/(2*x^2) end:
# S(0) is g.f. for Motzkin numbers A001006,
# S(1) is g.f. for this sequence,
# S(2) is g.f. for A004149, etc.

a[0]=1; a[n_Integer]:= a[n]= a[n-1]+Sum[a[k]*a[n-2-k], {k,n-2}]; Array[a, 35, 0]
CoefficientList[Series[(1-x+x^2-Sqrt[x^4-2x^3-x^2-2x+1])/(2x^2), {x,0,40}], x] (* Harvey P. Dale, May 09 2011 *)
a[n_]:= SeriesCoefficient[(1 -x +x^2 -Sqrt[1 -2x -x^2 -2x^3 +x^4])/(2x^2), {x, 0, n}]; (* Michael Somos, Jun 05 2014 *)
a[n_] := HypergeometricPFQ[{-n/2, (1-n)/2, (1-n)/2, 1-n/2}, {2, -n, -n + 1}, 16]; Array[a, 33, 0] (* Peter Luschny, Jan 25 2020 *)
Table[If[n==0,1, Sum[(Binomial[n-k,k+1]Binomial[n-k,k]/(n-k)), {k,0,n-1}]], {n,0,10}] (* Rigoberto Florez, Apr 17 2023 *)
CoefficientList[Nest[1+x(1-x) #+x^2 #^2 &, 1+O[x], 32], x](* Oliver Seipel, Dec 21 2024 *)

a(n):=coeff(taylor((1-x+x^2-sqrt(1-2*x-x^2-2*x^3+x^4))/(2*x^2),x,0,n),x, n); makelist(a(n),n,0,12); /* Emanuele Munarini, Jul 07 2001 */

{a(n) = polcoeff( (1 - x + x^2 - sqrt(1 - 2*x - x^2 + x^3 * (-2 + x + O(x^n)))) / 2, n + 2)}; /* Michael Somos, Jul 20 2003 */

a(n,m=1)=sum(k=0,n,sum(j=0,k,binomial(n-k+j+m,n-k)*m/(n-k+j+m)*binomial(n-k,k-j)*binomial(k-j,j))) \\ Paul D. Hanna, Jun 26 2009

{a(n)=polcoeff(1+x*exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^2*x^k)*x^m/m)+x*O(x^n)),n)} /* Paul D. Hanna, Mar 15 2011 */

{a(n)=local(A051292=1+(1-x^2)/sqrt((1-3*x+x^2)*(1+x+x^2) +x*O(x^n)));polcoeff(exp(sum(m=1,n,polcoeff(A051292,m)*x^m/m)+x*O(x^n)),n)} /* Paul D. Hanna, Mar 15 2011 */

{a(n) = my(A = 1 + O(x)); for(k=1, n, A = 1 - x / (x^2 - 1/A)); polcoeff( A, n)}; /* Michael Somos, Jun 05 2014 */

def A004148_list(prec):
    P = PowerSeriesRing(ZZ, 'x', prec)
    x = P.gen().O(prec)
    return ( (1-x+x^2 -sqrt(1-2*x-x^2-2*x^3+x^4))/(2*x^2) ).list()
A004148_list(35) # G. C. Greubel, Dec 30 2019

a(n+1) = a(n) + a(1)*a(n-2) + a(2)*a(n-3) + ... + a(n-1)*a(0).
G.f.: (1 - x + x^2 - sqrt(1 - 2*x - x^2 - 2*x^3 + x^4)) / (2*x^2). - Michael Somos, Jul 20 2003
G.f.: (1/z)*(1-C(-z/(1-3*z+z^2))), where C(z)=(1-sqrt(1-4*z))/(2*z) is the Catalan function. - Emeric Deutsch, Nov 19 2003
G.f.: 1 + F(x, x)/x, where F(x, t) is the g.f. of the Narayana numbers: xF^2 - (1-x-tx)F + tx = 0. - Emeric Deutsch, Nov 19 2003
G.f. A(x) satisfies the functional equation: x^2*A(x)^2 - (x^2 - x + 1)*A(x) + 1 = 0. - Michael Somos, Jul 20 2003
Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, Jul 20 2003
a(n) = A088518(2n) + A088518(2n+1) - A088518(2n+2). - Emeric Deutsch, Nov 19 2003
a(n) = Sum_{k=ceiling((n+1)/2)..n} (binomial(k, n-k)*binomial(k, n-k+1)/k) for n >= 1. - Emeric Deutsch, Nov 12 2003 This formula counts (i) disjoint-diagonal dissections by number of diagonals, (ii) peak-less Motzkin paths by number of up steps, (iii) UUU- and DDD-free Dyck paths by number of ascents. - David Callan, Mar 23 2004
a(n) = Sum_{k=0..floor(n/2)} A131198(n-k,k). - Philippe Deléham, Nov 06 2007
G.f.: 1/(1-x/(1-x^2/(1-x/(1-x^2/(1-x/(1-x^2/(1-x... (continued fraction). - Paul Barry, Dec 08 2008
G.f.: 1/(1-x/(1-x(x-1)-x/(1-x(x-1)-x/(1-x(x-1)-x/(1-... (continued fraction). - Paul Barry, May 16 2009
From Paul D. Hanna, Jun 26 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n, then
a(n,m) = Sum_{k=0..n} Sum_{j=0..k} C(n-k+j+m,n-k)*m/(n-k+j+m) * C(n-k,k-j)*C(k-j,j).
(End)
From Paul Barry, Mar 10 2010: (Start)
G.f.: (1/(1+x^2))*M(x/(1+x^2)), M(x) the g.f. of the Motzkin numbers A001006;
G.f.: 1/(1-x+x^2-x^2/(1-x+x^2-x^2/(1-x+x^2-x^2/(1-x+x^2-x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k,k)*A001006(n-2*k). (End)
G.f.: 1 + x*exp( Sum_{n>=1} (x^n/n)*(Sum_{k=0..n} C(n,k)^2*x^k) ). - Paul D. Hanna, Mar 15 2011
G.f.: exp( Sum_{n>=1} A051292(n)*x^n/n ), where A051292(n) is a Whitney number of level n. - Paul D. Hanna, Mar 15 2011
Let the g.f. be A(x), then B(x)=(1+x*A(x)) = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+x+x^2), B(x) = 1 +1*x + 1*x^2 +1*x^3 + 2*x^4 + 4*x^5 + ... is the g.f. of this sequence prepended with 1; more generally B(x) = C(x/(1+x+x^2)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
D-finite with recurrence: (n+2)*a(n) - (2n+1)*a(n-1) + (1-n)*a(n-2) + (5-2n)*a(n-3) + (n-4)*a(n-4) = 0. - R. J. Mathar, Dec 01 2011. This recurrence follows from the Wilf-Zeilberger (WZ) proof technique applied to Sum_{k=ceiling((n+1)/2)..n} (binomial(k,n-k) * binomial(k,n-k+1)/k). - T. Amdeberhan, Jul 23 2012
Given g.f. A(x), then B(x) = x * A(x) satisfies B(x) = x + x*B(x) / (1 - x*B(x)). - Michael Somos, Jun 05 2014
G.f.: 1 - x / (x^2 - 1 / (1 - x / (x^2 - 1 / (1 - x / (x^2 - ...))))). - Michael Somos, Jun 05 2014
0 = a(n)*(a(n+1) - 5*a(n+2) - 4*a(n+3) - 11*a(n+4) + 7*a(n+5)) + a(n+1)*(a(n+1) + 6*a(n+2) + 12*a(n+3) + 11*a(n+4) - 11*a(n+5)) + a(n+2)*(-a(n+2) - 7*a(n+3) + 12*a(n+4) - 4*a(n+5)) + a(n+3)*(-a(n+3) + 6*a(n+4) - 5*a(n+5)) + a(n+4)*(a(n+4) + a(n+5)) if n >= -1. - Michael Somos, Jun 05 2014
a(n) = hypergeom([-n/2, (1 - n)/2, (1 - n)/2, 1 - n/2], [2, -n, -n + 1], 16). - Peter Luschny, Jan 25 2020
a(n) = Sum_{k=0..n-1} binomial(n-k,k+1)*binomial(n-k,k)/(n-k) for n > 0. - Rigoberto Florez, Apr 17 2023
a(n) ~ 5^(1/4) * phi^(2*n + 2) / (2 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 05 2023

A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 63, 153, 376, 931, 2317, 5794, 14545, 36631, 92512, 234205, 594169, 1510192, 3844787, 9802895, 25027296, 63972861, 163701327, 419316330, 1075049011, 2758543201, 7083830648, 18204064403, 46812088751, 120452857976

Offset: 0

Emanuele Munarini

A Chebyshev transform of the central trinomial numbers A002426: image of 1/sqrt(1-2x-3x^2) under the mapping that takes g(x) to (1/(1+x^2))*g(x/(1+x^2)). - Paul Barry, Jan 31 2005
a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - Paul D. Hanna, May 22 2005
This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008
From Paul Barry, Mar 31 2010: (Start)
Apply the Riordan array (1/(1-x+x^2),x/(1-x+x^2)) to the aerated central binomial coefficients with g.f. 1/sqrt(1-4x^2).
Hankel transform is A174882. (End)
a(n) is the number of lattice paths in L[n]. The members of L[n] are lattice paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step h with weight 1, an (1,0)-step H with weight 2, a (1,1)-step U with weight 2, and a (1,-1)-step D with weight 1. The weight of a path is the sum of the weights of its steps. Example: a(3)=5 because we have hhh, hH, Hh, UD, and DU; a(4)=11 because we have hhhh, hhH, hHh, Hhh, HH, hUD, UhD, UDh, hDU, DhU, and DUh (see the Bona-Knopfmacher reference).
Apparently the number of peakless grand Motzkin paths of length n. - David Scambler, Jul 04 2013
A bijection between L[n] (as defined above) and peakless grand Motzkin paths of length n is now given in arXiv:2002.12874. - Sergi Elizalde, Jul 14 2021
a(n) is also the number of unimodal bargraphs with a centered maximum (i.e., whose column heights are weakly increasing in the left half and weakly decreasing in the right half) and semiperimeter n+1. - Sergi Elizalde, Jul 14 2021
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2}, and no nodes have the same weight as their parent node. - John Tyler Rascoe, Jun 10 2025

			a(3) = 5 because the ideals of size 3 of the fence F(6) = { x1 < x2 > x3 < x4 > x5 < x6 } are x1*x3*x5, x1*x2*x3, x3*x4*x5, x1*x5*x6, x3*x5*x6.

Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Cyril Banderier and Paweł Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. See Puzzle 3.1.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Miklós Bóna and Arnold Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2021.
Edyta Hetmaniok, Barbara Smoleń and Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_l^{1}(z).
Emanuele Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Jesús Sistos Barrón and Hua Wang, Balanced n-color compositions, Integers (2024) Vol. 24A, Art. No. A2. See pp. 4, 5, 8, 14.

Cf. main diagonal of A125250, column k=2 of A384747.

Cf. A051291, A051292, A078698, A107597, A185828 (log), A174882 (Hankel transf.).

seq( sum('binomial(i-k,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 5][n+1],
     ((2*n-1)*a(n-1)+(n-1)*a(n-2)+(2*n-3)*a(n-3)-(n-2)*a(n-4))/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 11 2016

Table[Sum[Binomial[n-k,k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Emanuele Munarini, Mar 01 2011; corrected by Harvey P. Dale, Sep 12 2012 *)
CoefficientList[Series[1/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)
a[n_] := HypergeometricPFQ[ {(1-n)/2, (1-n)/2, -n/2, -n/2}, {1, -n, -n}, 16]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 26 2013 *)

makelist(sum(binomial(n-k,k)^2,k,0,floor(n/2)),n,0,40);  /* Emanuele Munarini, Mar 01 2011 */

a(n)=polcoeff(1/sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)),n)

a(n)=sum(k=0,n,binomial(n-k,k)^2) /* Paul D. Hanna */

{a(n)=polcoeff( exp(sum(m=1,n, sum(k=0,m, binomial(2*m,2*k)*x^k) *x^m/m) +x*O(x^n)), n)}  /* Paul D. Hanna, Mar 18 2011 */

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, n, binomial(m+k, k)^2*x^k) * (1-x)^(2*m+1) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, n, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

from sympy import binomial
def a(n): return sum(binomial(n - k, k)**2 for k in range(n//2 + 1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Apr 18 2017

G.f.: 1/sqrt(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A002426(n-2k). - Paul Barry, Jan 31 2005
From Paul D. Hanna, May 22 2005: (Start)
a(n) = Sum_{k=0..n} C(n-k, k)^2.
Limit_{n->oo} a(n+1)/a(n) = (sqrt(5)+3)/2.
G.f.: 1/sqrt((1+x+x^2)*(1-3*x+x^2)). (End)
a(n) = Sum_{k=0..n} A049310(n, k)^2. - Philippe Deléham, Nov 21 2005
a(n) = Sum_{k=0..n} (C(k,k/2)*(1+(-1)^k)/2) * Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*((1+(-1)^(n-j))/2)*C(j,k). - Paul Barry, Mar 31 2010
G.f.: exp( Sum_{n>=1} (x^n/n)*Sum_{k=0..n} C(2n,2k)*x^k ). - Paul D. Hanna, Mar 18 2011
Logarithmic derivative equals A185828. - Paul D. Hanna, Mar 18 2011
D-finite with recurrence: n*a(n) - (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) + (n-2)*a(n-4) = 0. - R. J. Mathar, Dec 17 2011
The g.f. A(x) satisfies the differential equation (1-2*x-x^2-2*x^3+x^4)*A'(x) = (1+x+3*x^2-2*x^3)*A(x), from which the recurrence conjectured by Mathar follows. - Emanuele Munarini, Dec 18 2017
a(n) ~ phi^(2*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 05 2013, simplified Dec 18 2017
From Paul D. Hanna, Sep 05 2014: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k.
G.f.: Sum_{n>=0} x^n *[Sum_{k>=0} C(n+k,k)^2 * x^k] * (1-x)^(2*n+1).
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(End)

A110320 Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition).

Original entry on oeis.org

1, 2, 5, 13, 32, 80, 201, 505, 1273, 3217, 8146, 20668, 52531, 133726, 340909, 870213, 2223958, 5689807, 14571335, 37350585, 95821071, 246015677, 632088930, 1625119218, 4180845277, 10762096850, 27718352411, 71426753423, 184146711578

Offset: 1

Emeric Deutsch, Jul 19 2005

nonn

Antidiagonal sums of A132812. - Philippe Deléham, Jun 08 2013

			a(4)=13 because the 4 (=A004148(4)) RNA secondary structures of size 4, namely 1/2/3/4, 13/2/4, 14/2/3 and 1/24/3, have altogether 4+3+3+3=13 blocks.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A110319.

G:=1/2*(1-z-z^2)/z^2/(1-2*z-z^2-2*z^3+z^4)^(1/2)-1/2*1/(z^2): Gser:=series(G,z=0,37): seq(coeff(Gser,z^n),n=1..33);

Table[Sum[Binomial[n-j+1,j]Binomial[n-j+1,j-1],{j, 0, n}],{n,1,25}] (* Benedict W. J. Irwin, Sep 24 2016 *)

G.f.: (1-z-z^2)/(2*z^2*sqrt(1-2*z-z^2-2*z^3+z^4))-1/(2*z^2).
a(n) = Sum_{k=1..n} k*A110319(n,k).
Conjecture: a(n) = (A051292(n+2)-A051286(n+1))/2. - Gerald McGarvey, Jan 14 2007
a(n) = (A051286(n+2)-A051286(n+1)-A051286(n))/2. - Benedict W. J. Irwin, Sep 24 2016
a(n) ~ sqrt(4 + 9/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Sep 25 2016, equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(n-7)*a(n-3) +2*(2*n-3)*a(n-4) +(n-5)*a(n-5) +(-n+4)*a(n-6)=0. - R. J. Mathar, Feb 21 2020

A051291 Whitney number of level n of the lattice of the ideals of the fence of order 2 n + 1.

Original entry on oeis.org

1, 2, 3, 7, 17, 40, 97, 238, 587, 1458, 3640, 9124, 22951, 57904, 146461, 371281, 943045, 2399460, 6114555, 15603339, 39866932, 101976512, 261117378, 669239402, 1716737267, 4407306170, 11323050897, 29110603423, 74888578067

Offset: 0

Emanuele Munarini

nonn

This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008

			a(2) = 3 because the ideals of size 2 of the fence F(5) = { x1 < x2 > x3 < x4 > x5 } are x1x2, x1x3, x2x3.

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A051292.

G.f.: function = (1+2*t^2-t^3-(1-t)*sqrt(1-2*t-t^2-2*t^3+t^4))/(2*t*sqrt(1-2*t-t^2-2*t^3+t^4))

A098813 For a string of letters of length k, say abc...def, let f(k) be the string of length k-1 consisting of the adjacent pairs ab, bc, cd, ..., de, ef. Given n, let U be the string of length 2n consisting of n 1's followed by n 2's: 11...122...2. Then a(n) is the number of the C(2n,n) permutations V of U such that f(U) and f(V) agree in exactly one place.

Original entry on oeis.org

1, 1, 4, 19, 57, 178, 543, 1591, 4598, 13117, 36999, 103514, 287653, 794847, 2186054, 5988339, 16347999, 44497490, 120804023, 327217525, 884531586, 2386747391, 6429784509, 17296261734, 46465809007, 124678595953, 334173980818, 894778164125

Offset: 1

Zerinvary Lajos (with help from Miklos Kristof), Oct 07 2004

The number of V's such that f(U) and f(V) agree in no positions gives the A051292(n+1) sequence (Whitney numbers): 1, 4, 9, 21, 52, 127, 313, 778, 1941, 4863, 12228, 30817, ...

			For n=1, U = 12 and only one V, 12 is a 1-match, so a(1)=1.
For n=2, U = 1122, f(U) = 11,12,22 and only one V, 2121 is a 1-match, with f(v) = 21,12,21, so a(2)=1.
For n=3, U = 111222 and only the four V's 112212, 121122, 121221 and 221211 are 1-matches, so a(3)=4.

Cf. A051292.

with(combinat): for n from 1 to 10 do y:=0:B:=array: M:=[seq(11,i=1..n-1),seq(12,i=n),seq(22,i=n+1..2*n-1)]: S:=[seq(i,i=1..2*n)]: L:=choose(S,n): for j from 1 to binomial(2*n,n) do for k from 1 to 2*n-1 do if member(k,L[j]) then B[k]:=10 else B[k]:=20 end if: if member(k+1,L[j]) then B[k]:=B[k]+1 else B[k]:=B[k]+2 end if end do: x:=0: for l from 1 to 2*n-1 do if B[l]=M[l] then x:=x+1 end if end do: if x=1 then y:=y+1 end if end do: print(y) end do: # Miklos Kristof, Oct 07 2004

def find(bits_in, n0, n1, match):
    global count, U
    bitsleft = n0 + n1
    if bitsleft==0:
        if match:
            count += 1
    else:
        bitsleft -= 1
        if n0 > 0:
            bits_out = bits_in<<1
            new_match = (bits_out&3) == ((U >> bitsleft)&3)
            if not (match and new_match):
                find(bits_out, n0-1, n1, match or new_match)
        if n1 > 0:
            bits_out = (bits_in<<1)|1;
            new_match = (bits_out&3) == ((U >> bitsleft)&3)
            if not (match and new_match):
                find(bits_out, n0, n1-1, match or new_match)
def A098813(n):
    global count, U
    count = 0 ; U = (1<Bert Dobbelaere, Dec 23 2018

a(13)-a(15) from Ray Chandler, Oct 25 2004

a(16)-a(28) from Bert Dobbelaere, Dec 24 2018

A205810 Irregular triangle read by rows: Whitney numbers c_{n,k} (n >= 0, 0 <= k <= 2n) of Lucas lattices.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 4, 3, 3, 1, 1, 4, 6, 8, 9, 8, 6, 4, 1, 1, 5, 10, 15, 20, 21, 20, 15, 10, 5, 1, 1, 6, 15, 26, 39, 48, 52, 48, 39, 26, 15, 6, 1, 1, 7, 21, 42, 70, 98, 119, 127, 119, 98, 70, 42, 21, 7, 1

Offset: 0

N. J. A. Sloane, Jan 31 2012

			Triangle begins:
  1;
  1, 1,  1;
  1, 2,  1,  2,  1;
  1, 3,  3,  4,  3,  3,   1;
  1, 4,  6,  8,  9,  8,   6,   4,   1;
  1, 5, 10, 15, 20, 21,  20,  15,  10,  5,  1;
  1, 6, 15, 26, 39, 48,  52,  48,  39, 26, 15,  6,  1;
  1, 7, 21, 42, 70, 98, 119, 127, 119, 98, 70, 42, 21, 7, 1;
  ...

Mark E. AlSukaiti and Nafaa Chbili, Alexander and Jones Polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice, arXiv:2303.11398 [math.GT], 2023.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Main diagonal is A051292.

Cf. A079487, A097724.

c:= (n, k)-> `if`(k=2*n, 1, n*add(1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i), i=0..floor(k/2))): seq(seq(c(n, k), k=0..2*n), n=0..8);  # Leonid Bedratyuk, May 15 2018

T(n,k) = if (k==2*n, 1, n*sum(i=0, k\2, 1/(n-i)*binomial(n-i,n-k+i)*binomial(k-i-1,i)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 16 2018

c(n, k) = n*Sum_{i = 0..floor(k/2)} 1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i) for 0 <= k <= 2*n-1; c(n, 2*n) = 1. - Leonid Bedratyuk, May 15 2018
From Peter Bala, Jun 26 2025: (Start)
For n >= 1, the n-th row polynomial R(n, x) = x^n * t(n, 1 + x + 1/x), where t(n, x) = 2*Chebyshev_T(n, x/2) (AlSukaiti and Chbili, Proposition 2.1).
Conjecture: for n >= 1, t(n, x + 1) = Sum_{k = 0..n} c(n, n-k)*t(k, x) - c(n, n). Cf. A097724. (End)

A099793 Table read by rows where row n consists of 2n terms representing k-matches described in A098813.

Original entry on oeis.org

1, 1, 4, 1, 0, 1, 9, 4, 5, 1, 0, 1, 21, 19, 12, 9, 7, 1, 0, 1, 52, 57, 53, 47, 16, 16, 9, 1, 0, 1, 127, 178, 202, 154, 120, 85, 20, 25, 11, 1, 0, 1, 313, 543, 664, 636, 525, 310, 233, 133, 24, 36, 13, 1, 0, 1, 778, 1591, 2204, 2355, 2007, 1621, 1076, 549, 404, 191, 28, 49, 15

Offset: 1

Zerinvary Lajos and Ray Chandler, Oct 26 2004

For a string of letters of length k, say abc...def, let f(k) be the string of length k-1 consisting of the adjacent pairs ab, bc, cd, ..., de, ef. Given n, let U be the string of length 2n consisting of n 1's followed by n 2's: 11...122...2. Then T(n,k) = number of the C(2n,n) permutations V of U such that f(U) and f(V) agree in exactly k places, 0<=k<=2n-1.

			Table begins:
1: 1,1,
2: 4,1,0,1,
3: 9,4,5,1,0,1,
4: 21,19,12,9,7,1,0,1,
5: 52,57,53,47,16,16,9,1,0,1,
6: 127,178,202,154,120,85,20,25,11,1,0,1,
7: 313,543,664,636,525,310,233,133,24,36,13,1,0,1,
8: 778,1591,2204,2355,2007,1621,1076,549,404,191,28,49,15,1,0,1,
9: 1941,4598,7091,8223,8036,6722,4721,3457,1911,901,645,259,32,64,17,1,0,1

First column is A051292(n+1); second column is A098813; row sums = A000984.

A125226 Array, read by antidiagonals, where A(1,1) = A(1,2) = A(2,1) = A(2,2) = 1, A(n,k) = 0 if n<1 or k<1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 2, 2, 0, 0, 1, 4, 1, 0, 0, 0, 4, 4, 0, 0, 0, 0, 3, 9, 3, 0, 0, 0, 0, 1, 11, 11, 1, 0, 0, 0, 0, 0, 8, 21, 8, 0, 0, 0, 0, 0, 0, 4, 27, 27, 4, 0, 0, 0, 0, 0, 0, 1, 23, 52, 23, 1, 0, 0, 0, 0, 0, 0, 0, 13, 67, 67, 13, 0, 0, 0, 0, 0, 0, 0, 0, 5, 62, 127, 62, 5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 41

Offset: 1

Gerald McGarvey, Jan 14 2007

It appears that the main diagonal (1,1,4,9,21,...) is A051292 (Whitney number of level n of the lattice of the ideals of the crown of size 2 n). It appears that if b(n) = the n-th antidiagonal sum - A108014(n-1) then the sequence b(n) is the sequence 1,0,-2,0,1,0 repeated. n-th row sum = A052945(n).

			Array begins:
  1 1 0 0 0 0 0 ...
  1 1 2 1 0 0 0 ...
  0 2 4 4 3 1 0 ...
  ...

Cf. A051292, A052945, A108014.

A=matrix(22,22);A[1,1]=1;A[1,2]=1;A[2,1]=1;A[2,2]=1;A[3,2]=2;A[2,3]=2;A[2,4]=1;A[4,2]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))

A(1,1) = A(1,2) = A(2,1) = A(2,2) = 1, A(n,k) = 0 if n<1 or k<1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1)

cp's OEIS Frontend

A004148 Generalized Catalan numbers: a(n+1) = a(n) + Sum_{k=1..n-1} a(k)*a(n-1-k).

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A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

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A110320 Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition).

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A051291 Whitney number of level n of the lattice of the ideals of the fence of order 2 n + 1.

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A205810 Irregular triangle read by rows: Whitney numbers c_{n,k} (n >= 0, 0 <= k <= 2n) of Lucas lattices.

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