Search: id:A034807
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%I A034807
%S A034807 2,1,1,2,1,3,1,4,2,1,5,5,1,6,9,2,1,7,14,7,1,8,20,16,2,1,9,27,30,9,1,
%T A034807 10,35,50,25,2,1,11,44,77,55,11,1,12,54,112,105,36,2,1,13,65,156,182,
%U A034807 91,13,1,14,77,210,294,196,49,2,1,15,90,275,450,378,140,15,1,16,104
%N A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
%C A034807 These polynomials arise in the following setup. Suppose G and H are power 
               series satisfying G+H=G*H=1/x. Then G^n+H^n = (1/x^n)*L_n(-x).
%C A034807 Apart from signs, triangle of coefficients when 2cos(nt) is expanded 
               in terms of x=2cos(t). For example, 2cos(2t)=x^2-2, 2cos(3t)=x^3-3x 
               and 2cos(4t)=x^4-4x^2+2. - Anthony Robin (anthony_robin(AT)hotmail.com), 
               Jun 02 2004
%C A034807 Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.
%C A034807 Row n has 1+floor(n/2) terms. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 25 2004
%C A034807 T(n,k)=number of k-matchings of the cycle C_n (n>1). Example: T(6,2)=9 
               because the 2-matchings of the hexagon with edges a,b,c,d,e,f are 
               ac, ad, ae, bd, be, bf, ce, cf and df. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 25 2004
%C A034807 An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the 
               o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= 
               sum(T(n,k)*(-x)^k,k=0..floor(n/2)).
%C A034807 This triangle also supplies the absolute values of the coefficients in 
               the multiplication formulae for the Lucas numbers A000032.
%D A034807 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 148.
%D A034807 C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
%D A034807 T. J. Osler, Cardan polynomials and the reduction of radicals, Math. 
               Mag., 74 (No. 1, 2001), 26-32.
%D A034807 Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York, 
               etc.: John Wiley & Sons, 2001. (Chapter 13, "Pascal-like Triangles,
               " is devoted to the present triangle.)
%H A034807 T. D. Noe, <a href="b034807.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A034807 Moussa Benoumhani, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               A Sequence of Binomial Coefficients Related to Lucas and Fibonacci 
               Numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%H A034807 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LucasPolynomial.html">Lucas Polynomial</a>
%F A034807 Lucas polynomial coefficients: 1, -n, [n(n-3)]/2!, - [n(n-4)(n-5)]/3!, 
               [n(n-5)(n-6)(n-7)]/4!, - [n(n-6)(n-7)(n-8)(n-9)]/5!... - Herb Conn, 
               HCR 83, Box 93, Custer, SD 57730 and Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 28 2003
%F A034807 G.f.: (2-x)/(1-x-x^2*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 
               31 2003
%F A034807 T(n, k) = T(n-1, k)+T(n-2, k-1), n>1. T(n, 0) = 1, n>0. T(n, k) = binomial(n-k, 
               k)+binomial(n-k-1, k-1) = n*binomial(n-k-1, k-1)/k, 0< = 2*k< = n 
               except T(0, 0) = 2.
%F A034807 T(n,k)=(n*(n-1-k)!)/(k!*(n-2*k)!),n>0,k>=0. - Alexander Elkins (alexander_elkins(AT)hotmail.com), 
               Jun 09 2007
%e A034807 I have seen two versions of these polynomials: One version begins L_0 
               = 2, L_1 = 1, L_2 = 1+2*x, L_3 = 1+3*x, L_4 = 1+4*x+2*x^2, L_5 = 
               1+5*x+5*x^2, L_6 = 1+6*x+9*x^2+2*x^3, L_7 = 1+7*x+14*x^2+7*x^3, L_8 
               = 1+8*x+20*x^2+16*x^3+2*x^4, L_9 = 1+9*x+27*x^2+30*x^3+9*x^4, ...
%e A034807 The other version (probably the more official one) begins L_0(x) = 2, 
               L_1(x) = x, L_2(x) = 2+x^2, L_3(x) = 3*x+x^3, L_4(x) = 2+4*x^2+x^4, 
               tc
%e A034807 L5 = x^5 - 5x^3 + 5x = 1, -5, 5 = 1, -n, [n(n-3)]/2.
%e A034807 Comment from John Blythe Dobson, Oct 11 2007: Triangle begins:
%e A034807 2;
%e A034807 1;
%e A034807 1, 2;
%e A034807 1, 3;
%e A034807 1, 4, 2;
%e A034807 1, 5, 5;
%e A034807 1, 6, 9, 2;
%e A034807 1, 7, 14, 7;
%e A034807 1, 8, 20, 16, 2;
%e A034807 1, 9, 27, 30, 9;
%e A034807 1, 10, 35, 50, 25, 2;
%e A034807 1, 11, 44, 77, 55, 11;
%e A034807 1, 12, 54, 112, 105, 36, 2;
%e A034807 1, 13, 65, 156, 182, 91, 13;
%e A034807 1, 14, 77, 210, 294, 196, 49, 2;
%e A034807 1, 15, 90, 275, 450, 378, 140, 15;
%p A034807 T:=proc(n,k) if n=0 and k=0 then 2 elif k>floor(n/2) then 0 else n*binomial(n-k,
               k)/(n-k) fi end: for n from 0 to 15 do seq(T(n,k),k=0..floor(n/2)) 
               od; # yields sequence in triangular form (Deutsch)
%o A034807 (PARI) T(n,k)=if(k<0|2*k>n,0,binomial(n-k,k)+binomial(n-k-1,k-1)+(n==0&k==0))
%Y A034807 Row sums = A000032 (Lucas numbers). T(2n, n-1)=A000290(n), T(2n+1, n-1)=A000330(n), 
               T(2n, n-2)=A002415(n). T(n, k)=A029635(n-k, k), if n>0. See also 
               A061896.
%Y A034807 Cf. A114525
%Y A034807 Sequence in context: A055893 A050221 A113279 this_sequence A135062 A088428 
               A025838
%Y A034807 Adjacent sequences: A034804 A034805 A034806 this_sequence A034808 A034809 
               A034810
%K A034807 tabf,easy,nonn
%O A034807 0,1
%A A034807 N. J. A. Sloane (njas(AT)research.att.com).
%E A034807 Improved description, more terms, etc., from Michael Somos

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