cp's OEIS Frontend

Table starts

......1.........2.............5................14....................41

......3........14............95...............662..................4631

.....10.......122..........1985.............32414................529862

.....36......1094.........41675...........1588262..............60632429

....136......9842........875165..........77824814............6938214854

....528.....88574......18378455........3813415862..........793945203881

...2080....797162.....385947545......186857377214........90851753687090

...8256...7174454....8104898435.....9156011483462.....10396235291448605

..32896..64570082..170202867125...448644562689614...1189649113515482414

.131328.581130734.3574260209615.21983583571791062.136132453105625552657

Table starts

......1........2............5..............14.................41

......3.......11...........65.............386...............2315

.....10.......74..........941...........11486.............141566

.....36......515........13721..........342626............8714705

....136.....3602.......200165........10221326..........537122150

....528....25211......2920145.......304926626........33113065637

...2080...176474.....42601181......9096692126......2041493495546

...8256..1235315....621496841....271376130626....125863931140721

..32896..8647202...9066845525...8095800458126...7759890074654654

.131328.60530411.132273701825.241517133090626.478420800866866973

Table starts

....1.....2.......5.......14.........41..........122............365

....2.....9......54......324.......1944........11664..........69984

....5....24.....128......688.......3728........20224.........109760

...14....64.....396.....2564......17036.......114184.........767400

...41...172....1440....13156.....126420......1224088.......11894712

..122...476....5676....73012....1020324.....14005180......194398028

..365..1320...22844...409728....8391716....161725644.....3224773976

.1094..3672...93968..2315520...70863268...1922952988....56045462432

.3281.10220..389820.13124196..603245904..22903224024...981633748272

.9842.28472.1626348.74374784.5149743348.271600215464.17187870741644

Table starts

...1....2.....5......14.......41.......122.........365.........1094

...2....9....54.....324.....1944.....11664.......69984.......419904

...3...16....84.....444.....2344.....12376.......65344.......345008

...6...31...178....1011.....5758.....32771......186538......1061795

..12...63...394....2404....14884.....91849......566331......3495079

..24..129..1017....8122....65045....518049.....4142271.....33102260

..48..260..2645...27373...283169...2940064....30551417....317215507

..96..534..6980...94065..1266905..17159883...232200615...3146508594

.192.1083.18464..323306..5656712.100110527..1766745035..31272546310

.384.2210.48959.1109170.25396322.587134035.13518543301.312946900074

Table starts

....1.....1.......2........4..........8...........16............32

....2.....4......12.......36........108..........324...........972

....5....11......45......173........693.........2765.........11061

...14....31.....189.....1017.......5909........33461........191289

...41....88.....805.....5965......50949.......408105.......3363533

..122...250....3437....34865.....442001......4988145......59728757

..365...710...14693...203933....3861469.....61239977....1073114625

.1094..2016...62829..1192701...33851605....752660245...19398127957

.3281..5724..268677..6974781..297360321...9254592049..352134188049

.9842.16252.1148973.40786925.2615328377.113817204341.6411366745009

Number of permutations in S_n avoiding both 132 and 123...k.

T(n,k) = number of rooted ordered trees on n nodes of depth <= k. Also, T(n,k) = number of {1,-1} sequences of length 2n summing to 0 with all partial sums are >=0 and <= k. Also, T(n,k) = number of closed walks of length 2n on a path of k nodes starting from (and ending at) a node of degree 1. - Mitch Harris, Mar 06 2004

Also T(n,k) = k-th coefficient in expansion of the rational function R(n), where R(1) = 1, R(n+1) = 1/(1-x*R(n)), which means also that lim(n->inf,R(n)) = g.f. of Catalan numbers (A000108) wherever it has real value (see Mansour article). - Clark Kimberling and Ralf Stephan, May 26 2004

Row n of the array gives Taylor series expansion of F_n(t)/F_{n+1}(t), where F_n(t) are the Fibonacci polynomials defined in A259475 [Kreweras, 1970]. - N. J. A. Sloane, Jul 03 2015

A tetrahedron has 4 faces. Cut every corner so that we get triangular faces; the resulting polyhedron has 8 faces. Repeating this procedure gives polyhedra with 4, 8, 20, 56, etc. faces.

Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x).

Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006

Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009

Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012

Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014

The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015

The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020

The n-th row of the triangle is obtained by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probabilities of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms of the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k-1) as the (2k-1)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k-1). - Luca Onnis, Oct 29 2023

Table starts

.....1......1.........1..........4...........5...........14...........17

.....2......1.........5.........10..........25..........128..........292

.....5......5........11.........56.........134.........1500.........3999

....14.....18.......120........435........3516........25441.......212957

....41.....59.......500.......4790.......30766.......543064......4102865

...122....200......5210......50535......881766.....14120189....278335019

...365....820.....20889.....547563.....9004283....406233879...7240361264

..1094...3014....229894....5815939...287554360..12231597130.617491399888

..3281..11364....913405...63130202..2996904309.379056994144

..9842..42748..10427244..676152734.97314847813

.29525.164652..40896923.7328771224

.88574.627034.479942958