%I A008459
%S A008459 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 A008459 36,1,1,49,441,1225,1225,441,49,1,1,64,784,3136,4900,3136,784,64,1,1,
%U A008459 81,1296,7056,15876,15876,7056,1296,81,1,1,100,2025,14400,44100,63504
%N A008459 Square the entries of Pascal's triangle.
%C A008459 Number of lattice paths from (0,0) to (n,n) with steps (1,0) and (0,1), 
               having k right turns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 23 2003
%C A008459 Product of A007318 and A105868. - Paul Barry (pbarry(AT)wit.ie), Nov 
               15 2005
%C A008459 Number of partitions that fit in an n X n box with Durfee square k. - 
               Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 20 2006
%C A008459 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: 
               (Start)
%C A008459 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 A008459 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 A008459 (End)
%D A008459 J. Riordan, An introduction to combinatorial analysis, Dover Publications, 
               Mineola, NY, 2002, page 191, Problem 15. MR1949650
%H A008459 A. Necer, <a href="http://www.emis.de/journals/JTNB/1997-2/jtnb9-2_english.html#jourelec">
               Series formelles et produit de Hadamard</a>
%H A008459 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> [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]
%H A008459 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
               Root systems and generalized associahedra</a>, Lecture notes for 
               IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 
               23 2008]
%F A008459 Cf. A007318, A055133.
%F A008459 E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Nov 17 2003
%F A008459 G.f.: 1/sqrt(1-2*y-2*x*y+y^2-2*x*y^2+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 (deutsch(AT)duke.poly.edu), Nov 23 2003
%F A008459 G.f. for column k is sum(C(k, j)^2*x^(k+j), j, 0, k)/(1-x)^(2k+1). - 
               Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
%F A008459 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 (pbarry(AT)wit.ie), Nov 
               19 2005
%F A008459 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. [From Peter Bala 
               (pbala(AT)toucansurf.com), Oct 23 2008]
%e A008459 1; 1,1; 1,4,1; 1,9,9,1; 1,16,36,16,1; ...
%p A008459 binomial(n,k)^2;
%o A008459 (PARI) T(n,k)=if(k<0|k>n,0,binomial(n,k)^2)
%Y A008459 Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
%Y A008459 Cf. A116647.
%Y A008459 A001263, A086645, A063007, A108558, A108625(Hilbert transform), A145903. 
               [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]
%Y A008459 Sequence in context: A152237 A082043 A124216 this_sequence A157192 A154982 
               A146767
%Y A008459 Adjacent sequences: A008456 A008457 A008458 this_sequence A008460 A008461 
               A008462
%K A008459 nonn,tabl
%O A008459 0,5
%A A008459 N. J. A. Sloane (njas(AT)research.att.com).