This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A054391 #87 Aug 14 2025 06:06:43
%S A054391 1,1,2,5,14,41,123,374,1147,3538,10958,34042,105997,330632,1032781,
%T A054391 3229714,10109310,31667245,99260192,311294876,976709394,3065676758,
%U A054391 9625674442,30231524869,94972205349,298419158008,937861780439,2947969125284,9267666915326
%N A054391 Number of permutations with certain forbidden subsequences.
%C A054391 Hankel transform is [1,1,1,...] = A000012. - _Paul Barry_, Jan 19 2009
%C A054391 The inverse Motzkin transform apparently yields 1 followed by A000930, which implies a generating function g(x)=1+z/(1-z-z^3) where z=x*A001006(x). - _R. J. Mathar_, Jul 07 2009
%C A054391 It appears that the infinite set of interpolated sequences between the Motzkin and the Catalan can be generated with a succession of INVERT transforms, given each sequence has two leading 1's. Also, the N-th sequence in the set starting with (N=1, A001006) can be generated from a production matrix of the form "M" in the formula section, such that the main diagonal is (N leading 1's, 0, 0, 0, ...). M with a diagonal of (1, 0, 0, 0, ...) generates A001006, while M with a main diagonal of all 1's is the production matrix for A000108. - _Gary W. Adamson_, Jul 29 2011
%C A054391 From _Gus Wiseman_, Jun 22 2019: (Start)
%C A054391 Conjecture: Also the number of non-capturing set partitions of {1..n}. A set partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting. The a(0) = 1 through a(4) = 14 non-capturing set partitions are:
%C A054391 {} {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
%C A054391 {{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
%C A054391 {{1,2},{3}} {{1,2},{3,4}}
%C A054391 {{1,3},{2}} {{1,2,3},{4}}
%C A054391 {{1},{2},{3}} {{1,2,4},{3}}
%C A054391 {{1,3},{2,4}}
%C A054391 {{1,3,4},{2}}
%C A054391 {{1},{2},{3,4}}
%C A054391 {{1},{2,3},{4}}
%C A054391 {{1,2},{3},{4}}
%C A054391 {{1},{2,4},{3}}
%C A054391 {{1,3},{2},{4}}
%C A054391 {{1,4},{2},{3}}
%C A054391 {{1},{2},{3},{4}}
%C A054391 Cf. A000108, A000110, A058681, A326212, A326237, A326243, A326244, A326249, A326255.
%C A054391 (End)
%C A054391 The above conjecture is true: A partition is non-capturing iff its representation in canonical sequential form avoids the patterns 1221 and 2112. In the context of these partition representations, the pattern 2112 is equivalent to the pattern 12112. Partitions whose canonical sequence form avoid 1221 and 12112 are one of the classes that are handled in the Mansour/Shattuck "Pattern Avoiding Partitions,..." paper. It shows that they are counted by this sequence. - _Christian Sievers_, Oct 29 2024
%H A054391 G. C. Greubel, <a href="/A054391/b054391.txt">Table of n, a(n) for n = 0..1000</a>
%H A054391 E. Barcucci et al., <a href="http://dx.doi.org/10.1016/S0012-365X(99)00254-X">From Motzkin to Catalan Permutations</a>, Discr. Math., 217 (2000), 33-49.
%H A054391 Jean-Luc Baril and Sergey Kirgizov, <a href="https://arxiv.org/abs/2104.01186">Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths</a>, arXiv:2104.01186 [math.CO], 2021.
%H A054391 Paul Barry, <a href="http://arxiv.org/abs/1205.2565">On sequences with {-1, 0, 1} Hankel transforms</a>, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
%H A054391 Petr Gregor, Torsten Mütze, and Namrata, <a href="https://arxiv.org/abs/2306.08420">Combinatorial generation via permutation languages. VI. Binary trees</a>, arXiv:2306.08420 [cs.DM], 2023.
%H A054391 Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H A054391 J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5
%H A054391 Toufik Mansour and Mark Shattuck, <a href="https://digitalcommons.pvamu.edu/aam/vol6/iss2/1">Pattern Avoiding Partitions, Sequence A054391 and the Kernel Method</a>, Applications and Applied Mathematics, Vol. 6, Issue 2 (December 2011), pp. 397-411.
%H A054391 T. Mansour and M. Shattuck, <a href="https://web.archive.org/web/20210613072823/http://puma.dimai.unifi.it/22_2/mansour_shattuck.pdf">Restricted partitions and generalized Catalan numbers</a>, PU. M. A., Vol. (2011), No. 2, pp. 239-251. - From _N. J. A. Sloane_, Oct 13 2012
%H A054391 Eric Marberg, <a href="http://arxiv.org/abs/1203.5738">Crossings and nestings in colored set partitions</a>, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
%F A054391 G.f.: 1 - 2*x^2 / (2*x^2 - 3*x + 1 - sqrt(1 - 2*x - 3*x^2)). - Mansour and Shattuck
%F A054391 G.f.: 1/(1-x-x^2/(1-2x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction) (conjecture). - _Paul Barry_, Jan 19 2009
%F A054391 From _Gary W. Adamson_, Jul 29 2011: (Start)
%F A054391 a(n) = upper left term of M^n, a(n+1) = sum of top row terms of M^n; M = an infinite square production matrix as follows with a main diagonal of (1, 1, 1, 0, 0, 0, ...):
%F A054391 1, 1, 0, 0, 0, 0, ...
%F A054391 1, 1, 1, 0, 0, 0, ...
%F A054391 1, 1, 1, 1, 0, 0, ...
%F A054391 1, 1, 1, 0, 1, 0, ...
%F A054391 1, 1, 1, 1, 0, 1, ...
%F A054391 1, 1, 1, 1, 1, 0, ...
%F A054391 ... (End)
%F A054391 a(n) = Sum_{k=1..n-1} (sum(l=1..k, (binomial(k,l)*l*sum(j=0..n+l-k-1, binomial(j,1-n-2*l+k+2*j)*binomial(n-1+l-k,j)))/(n+l-k-1))) + 1. - _Vladimir Kruchinin_, Oct 31 2011
%F A054391 D-finite with recurrence (-n+1)*a(n) + 3*(2*n-3)*a(n-1) + (-8*n+11)*a(n-2) + (-5*n+32)*a(n-3) + (7*n-31)*a(n-4) + 3*(-n+4)*a(n-5)= 0. - _R. J. Mathar_, Nov 26 2012
%F A054391 G.f.: 1 - x*(2*x^2-3*x+1 + 1/G(0))/(2*(x^3-3*x^2+4*x-1)), 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_, Jun 29 2013
%e A054391 a(4) = 14, a(5) = 41 since the top row of M^4 = (14, 14, 9, 3, 1), with 41 = (14 + 14 + 9 + 3 + 1).
%p A054391 c := x->(1-sqrt(1-4*x))/(2*x); a := (x,j)->(x)/((1-4*x)*(c(x))^2*(1-c(x))^(j))*(-x^2*(c(x))^2*(1-c(x))*(x^2*(c(x))^4)^(j)-(1-3*x-2*x^2)*(c(x))^2*(x*(c(x))^2)^(j)+x);
%p A054391 b := (x,j)->1+(1)/((1-4*x)*c(x)*(1-c(x))^(j))*(-2*x^3*(c(x))^2*(x^2*(c(x))^4)^(j)+(1-3*x-2*x^2)*c(x)*(x*(c(x))^2)^(j)-2*x^2);
%p A054391 co := (x,j)->(1)/((1-4*x)*(1-c(x))^(j))*(x^2*(x^2*(c(x))^4)^(j)-(1-3*x-2*x^2)*(x*(c(x))^2)^(j)+x^2);
%p A054391 s := (x,j)->(1-b(x,j)+(-1)^j*sqrt((1-b(x,j))^2-4*a(x,j)*co(x,j)))/(2*a(x,j)); j := 3; series(s(x,j),x=0..60); od; # j=1,2,3,... inf gives A001006, A005773, A054391, A054392, ..., A000108
%t A054391 CoefficientList[Series[1 - 2*x^2/(2*x^2 - 3*x + 1 - Sqrt[1 - 2*x - 3*x^2]), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 27 2017 *)
%o A054391 (Maxima) a(n):=sum((sum((binomial(k,l)*l*sum(binomial(j,1-n-2*l+k+2*j)*binomial(n-1+l-k,j),j,0,n+l-k-1))/(n+l-k-1),l,1,k)),k,1,n-1)+1; /* _Vladimir Kruchinin_, Oct 31 2011 */
%o A054391 (PARI) x='x+O('x^66); gf=1-2*x^2/(2*x^2-3*x+1-sqrt(1-2*x-3*x^2)); Vec(gf) \\ _Joerg Arndt_, Jun 29 2013
%Y A054391 Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108). Cf. A005773, A054392, ...
%Y A054391 Binomial transform of A224747.
%K A054391 nonn
%O A054391 0,3
%A A054391 _N. J. A. Sloane_, Elisa Pergola (elisa(AT)dsi.unifi.it), May 21 2000