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%I #190 Mar 29 2024 10:44:50
%S 1,1,1,1,4,1,1,9,9,1,1,16,36,16,1,1,25,100,100,25,1,1,36,225,400,225,
%T 36,1,1,49,441,1225,1225,441,49,1,1,64,784,3136,4900,3136,784,64,1,1,
%U 81,1296,7056,15876,15876,7056,1296,81,1,1,100,2025,14400,44100,63504,44100,14400,2025,100,1
%N Square the entries of Pascal's triangle.
%C Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - _Emeric Deutsch_, Nov 23 2003
%C Product of A007318 and A105868. - _Paul Barry_, Nov 15 2005
%C Number of partitions that fit in an n X n box with Durfee square k. - _Franklin T. Adams-Watters_, Feb 20 2006
%C From _Peter Bala_, Oct 23 2008: (Start)
%C Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).
%C Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
%C (End)
%C The n-th row consists of the coefficients of the polynomial P_n(t) = int((1 + t^2 - 2*t*cos(s))^n, s = 0..2*Pi)/Pi/2. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - _Theodore Kolokolnikov_, Oct 26 2010
%C Let E(y) = sum {n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - _Peter Bala_, Jul 24 2013
%C From _Colin Defant_, Sep 16 2018: (Start)
%C Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.
%C T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)
%C The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - _Günter Rote_, Feb 27 2019
%C Let P be the poset [n] X [n] ordered by the product order. T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - _Geoffrey Critzer_, Mar 28 2020
%D T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.
%D J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
%D P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see p. 97) [From _Tom Copeland_, Sep 09 2010, vol number corrected Sep 10 2010]
%H G. C. Greubel, <a href="/A008459/b008459.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, <a href="https://arxiv.org/abs/2010.11157">Refined Catalan and Narayana cyclic sieving</a>, arXiv:2010.11157 [math.CO], 2020.
%H N. Alexeev and A. Tikhomirov, <a href="http://arxiv.org/abs/1501.04615">Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials</a>, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
%H F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, <a href="http://arxiv.org/abs/0809.5123">Root polytopes and growth series of root lattices</a>, arXiv:0809.5123 [math.CO], 2008.
%H Peter Bala, <a href="/A008459/a008459.pdf">A commutative diagram of triangular arrays</a>
%H E. Barcucci, A. Frosini and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.disc.2005.01.006">On directed-convex polyominoes in a rectangle</a>, Discr. Math., 298 (2005), 62-78.
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
%H Carl M. Bender and Gerald V. Dunne, <a href="http://dx.doi.org/10.1063/1.527869 ">Polynomials and operator orderings</a>, J. Math. Phys. 29 (1988), 1727-1731.
%H Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, <a href="https://arxiv.org/abs/1903.01095">The Number of Convex Polyominoes with Given Height and Width</a>, arXiv:1903.01095 [math.CO], 2019.
%H John H. Conway and N. J. A. Sloane, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.9899">Low-dimensional lattices. VII Coordination sequences</a>, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
%H R. Cori and G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
%H R. Cori and G. Hetyei, <a href="https://doi.org/10.46298/dmtcs.2404">How to count genus one partitions</a>, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
%H Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H Sergey Fomin and Nathan Reading, <a href="https://arxiv.org/abs/math/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005, 2008. [From _Peter Bala_, Oct 23 2008]
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Abdelkader Necer, <a href="https://doi.org/10.5802/jtnb.205">Séries formelles et produit de Hadamard</a>, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
%H Weiping Wang and Tianming Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%H Yi Wang and Arthur L.B. Yang, <a href="https://arxiv.org/abs/1702.07822">Total positivity of Narayana matrices</a>, arXiv:1702.07822 [math.CO], 2017.
%H Harold R. L. Yang and Philip B. Zhang, <a href="https://arxiv.org/abs/2403.15058">Stable multivariate Narayana polynomials and labeled plane trees</a>, arXiv:2403.15058 [math.CO], 2024. See p. 2.
%F T(n,k) = A007318(n,k)^2. - _Sean A. Irvine_, Mar 29 2018
%F E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - _Vladeta Jovovic_, Nov 17 2003
%F G.f.: 1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - _Emeric Deutsch_, Nov 23 2003 [The original version of the bivariate g.f. has been modified with the roles of x and y interchanged so that now x corresponds to n and y to k. - _Petros Hadjicostas_, Oct 22 2017]
%F G.f. for column k is sum(C(k, j)^2*x^(k+j), j, 0, k)/(1-x)^(2k+1). - _Paul Barry_, Nov 15 2005
%F Column k has g.f. x^k*Legendre_P(k, (1+x)/(1-x))/(1-x)^(k+1) = x^k*sum{j = 0..k, C(k, j)^2*x^j}/(1-x)^(2k+1). - _Paul Barry_, Nov 19 2005
%F Let E be the operator D*x*D, where D denotes the derivative operator d/dx. Then (1/n!^2) * E^n(1/(1-x)) = (row n generating polynomial)/(1-x)^(2n+1) = Sum_{k = 0..inf} binomial(n + k, k)^2*x^k. For example, when n = 3 we have (1/3!)^2*E^3(1/(1-x)) = (1 + 9*x + 9*x^2 + x^3)/(1-x)^7 = (1/3!)^2 * Sum_{k = 0..inf} [(k+1)*(k+2)*(k+3)]^2*x^k. - _Peter Bala_, Oct 23 2008
%F G.f.: A(x, y) = Sum_{n >= 0} (2n)!/n!^2 * x^(2n)*y^n/(1-x-x*y)^(2n+1). - _Paul D. Hanna_, Oct 31 2010
%F From _Peter Bala_, Jul 24 2013: (Start)
%F Let E(y) = sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Generating function: E(y)*E(x*y) = 1 + (1 + x)*y + (1 + 4*x + x^2)*y^2/2!^2 + (1 + 9*x + 9*x^2 + x^3)*y^3/3!^2 + .... Cf. the unsigned version of A021009 with generating function exp(y)*E(x*y).
%F The n-th power of this array has the generating function E(y)^n*E(x*y). In particular, the matrix inverse A055133 has the generating function E(x*y)/E(y). (End)
%F T(n,k) = T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1), T(n,0)=T(n,n)=1. - _Vladimir Kruchinin_, Oct 18 2014
%F Observe that the recurrence T(n,k) = T(n-1,k)*(n+k)/(n-k) - T(n-1,k-1), for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1 gives Pascal's triangle A007318. - _Peter Bala_, Dec 21 2014
%F n-th row polynomial R(n,x) = [z^n] (1 + (1 + x)*z + x*z^2)^n. Note that 1/n*[z^(n-1)] (1 + (1 + x)*z + x*z^2)^n gives the row polynomials of A001263. - _Peter Bala_, Jun 24 2015
%F Binomial transform of A105868. If G(x,t) = 1/sqrt(1 - 2*(1 + t)*x + (1 - t)^2*x^2) denotes the o.g.f. of this array then 1 + x*d/dx(log(G(x,t)) = 1 + (1 + t)*x + (1 + 6*t + t^2)*x^2 + ... is the o.g.f. for A086645. - _Peter Bala_, Sep 06 2015
%F T(n,k) = Sum_{i=0..n} C(n-i,k)*C(n,i)*C(n+i,i)*(-1)^(n-i-k). - _Vladimir Kruchinin_, Jan 14 2018
%F G.f. satisfies A(x,y) = x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2). - _Vladimir Kruchinin_, Oct 23 2020
%e Pascal's triangle begins
%e 1
%e 1 1
%e 1 2 1
%e 1 3 3 1
%e 1 4 6 4 1
%e 1 5 10 10 5 1
%e 1 6 15 20 15 6 1
%e 1 7 21 35 35 21 7 1
%e ...
%e so the present triangle begins
%e 1
%e 1 1
%e 1 4 1
%e 1 9 9 1
%e 1 16 36 16 1
%e 1 25 100 100 25 1
%e 1 36 225 400 225 36 1
%e 1 49 441 1225 1225 441 49 1
%e ...
%p seq(seq(binomial(n, k)^2, k=0..n), n=0..10);
%t Table[Binomial[n, k]^2, {n, 0, 11}, {k, 0, n}]//Flatten (* _Alonso del Arte_, Dec 08 2013 *)
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, binomial(n, k)^2)}; /* _Michael Somos_, May 03 2004 */
%o (PARI) {T(n,k)=polcoeff(polcoeff(sum(m=0,n,(2*m)!/m!^2*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(2*m+1)),n,x),k,y)} \\ _Paul D. Hanna_, Oct 31 2010
%o (Maxima) create_list(binomial(n,k)^2,n,0,12,k,0,n); \\ _Emanuele Munarini_, Mar 11 2011
%o (Maxima) T(n,k):=if n=k then 1 else if k=0 then 1 else T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1); /* _Vladimir Kruchinin_, Oct 18 2014 */
%o (Magma) /* As triangle */ [[Binomial(n, k)^2: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Dec 15 2016
%o (GAP) Flat(List([0..10],n->List([0..n],k->Binomial(n,k)^2))); # _Muniru A Asiru_, Mar 30 2018
%o (Maxima)
%o A(x,y):=1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2);
%o taylor(x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2),x,0,7,y,0,7); /* _Vladimir Kruchinin_, Oct 23 2020 */
%Y Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
%Y Family of polynomials (see A062145): this sequence (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
%Y Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625 (Hilbert transform), A145903, A181543, A086645 (logarithmic derivative), A105868 (inverse binomial transform), A093118.
%K nonn,tabl,easy
%O 0,5
%A _N. J. A. Sloane_
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