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A008459
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Square the entries of Pascal's triangle.
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43
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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, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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
Product of A007318 and A105868. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
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
Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: (Start)
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).
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.
(End)
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. [From Theodore Kolokolnikov (tkolokol(AT)gmail.com), Oct 26 2010]
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REFERENCES
| J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see pg. 97) [From Tom Copeland (tcjpn(AT)msn.com), Sep 09 2010, vol number corrected Sep 10 2010]
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LINKS
| F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]
A. Necer, Series formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
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FORMULA
| E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 17 2003
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
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
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
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]
G.f.: A(x,y) = Sum_{n>=0} (2n)!/n!^2 * x^(2n)*y^n/(1-x-x*y)^(2n+1). [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 31 2010]
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EXAMPLE
| 1; 1,1; 1,4,1; 1,9,9,1; 1,16,36,16,1; ...
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MAPLE
| binomial(n, k)^2;
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PROG
| (PARI) T(n, k)=if(k<0|k>n, 0, binomial(n, k)^2)
(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 31 2010]
(Maxima) create_list(binomial(n, k)^2, n, 0, 12, k, 0, n); [Emanuele Munarini, Mar 11 2011]
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CROSSREFS
| Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625(Hilbert transform), A145903, A181543.
Sequence in context: A177944 A174006 A124216 * A180960 A157192 A154982
Adjacent sequences: A008456 A008457 A008458 * A008460 A008461 A008462
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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