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A004148 Generalized Catalan numbers: a(n+1) = a(n) + Sum_{k=1..n-1} a(k)*a(n-1-k).
(Formerly M1141)
189
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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
REFERENCES
Cameron, Naiomi, and Everett Sullivan. "Peakless Motzkin paths with marked level steps at fixed height." Discrete Mathematics 344.1 (2021): 112154.
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).
LINKS
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, 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, 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.
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.
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.
Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.
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.
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.
E. Munarini and N. Z. 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.
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
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.
FORMULA
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
EXAMPLE
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
MAPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) {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 */
(PARI) 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
(PARI) {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 */
(PARI) {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 */
(Maxima) 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
(Haskell)
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
(PARI) {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 */
(Magma) R<x>:=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
(Sage)
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
CROSSREFS
Second row of A064645.
Cf. A046980 (Hankel transform).
Sequence in context: A025241 A292461 A203019 * A292460 A085022 A324936
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved

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Last modified March 18 22:22 EDT 2024. Contains 370951 sequences. (Running on oeis4.)