Search: id:A029635
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%I A029635
%S A029635 1,1,2,1,3,2,1,4,5,2,1,5,9,7,2,1,6,14,16,9,2,1,7,20,30,25,11,2,1,8,27,
               50,
%T A029635 55,36,13,2,1,9,35,77,105,91,49,15,2,1,10,44,112,182,196,140,64,17,2,1,
%U A029635 11,54,156,294,378,336,204,81,19,2,1,12,65,210,450,672,714,540,285,100
%N A029635 The (1,2)-Pascal triangle (or Lucas triangle) read by rows.
%C A029635 Dropping the first term and changing the boundary conditions to T(n,1)=n, 
               T(n,n-1)=2 (n>=2), T(n,n)=1 yields the number of nonterminal symbols 
               (which generate strings of length k) in a certain context-free grammar 
               in Chomsky normal form that generates all permutations of n symbols. 
               Summation over k (1<=k<=n) results in A003945. For the number of 
               productions of this grammar: see A090327. Example: 1; 2, 1; 3, 2, 
               1; 4, 5, 2, 1; 5, 9, 7, 2, 1; 6, 14, 16, 9, 2, 1; In addition to 
               the example of A090327 we have T(3,3)=#{S}=1, T(3,2)=#{D,E}=2 and 
               T(3,1)=#{A,B,C}=3. - Peter R. J. Asveld (infprja(AT)cs.utwente.nl), 
               Jan 29 2004
%D A029635 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), 
               FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published 
               by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; 
               see p. 25.
%D A029635 P. R. J. Asveld, Generating all permutations by context-free grammars 
               in Chomsky normal form, Theoretical Computer Science 354 (2006) 118-130.
%F A029635 T(n, k) =T(n-1, k-1)+T(n-1, k) =C(n, k)+C(n-1, k-1) =C(n, k)*(n+k)/n 
               =A007318(n, k)+A007318(n-1, k-1) =A061896(n+k, k) but with T(0, 0)=1 
               and T(1, 1)=2. Row sum is floor[3^2(n-1)] i.e. A003945. - Henry Bottomley 
               (se16(AT)btinternet.com), Apr 26 2002
%F A029635 G.f.: (1+xy)/(1-x-xy). - Michael Somos, Jul 15 2003
%F A029635 G.f. for n-th row: (x+2*y)*(x+y)^(n-1).
%F A029635 O.g.f. for row n: (1+x)/(1-x)^(n+1). The entries in row n are the nonzero 
               entries in column n of A053120 divided by 2^(n-1). - Peter Bala (pbala(AT)toucansurf.com), 
               Aug 14 2008
%F A029635 T(2n,n)-T(2n,n+1)= Catalan(n)= A000108(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 19 2009]
%e A029635 1; 1 2; 1 3 2; 1 4 5 2; 1 5 9 7 2; ...
%e A029635 Comment from Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008: Read 
               as a square, the array begins
%e A029635 ===================================
%e A029635 n\k|...0...1....2.....3.....4.....5
%e A029635 ===================================
%e A029635 0..|...1...2....2.....2.....2.....2 A040000
%e A029635 1..|...1...3....5.....7.....9....11 A005408
%e A029635 2..|...1...4....9....16....25....36 A000290
%e A029635 3..|...1...5...14....30....55....91 A000330
%e A029635 4..|...1...6...20....50...105...196 A002415
%e A029635 5..|...1...7...27....77...182...378 A005585
%e A029635 6..|...1...8...35...112...294...672 A040977
%e A029635 ...
%o A029635 (PARI) T(n,k)=if(k<0|k>n,0,binomial(n,k)+binomial(n-1,k-1))
%Y A029635 Cf. A007318, A034807, A061896.
%Y A029635 Sums along ascending diagonals give Lucas numbers, n>0.
%Y A029635 Cf. A090327, A003945.
%Y A029635 Sequence in context: A049280 A108786 A008315 this_sequence A104741 A167237 
               A089353
%Y A029635 Adjacent sequences: A029632 A029633 A029634 this_sequence A029636 A029637 
               A029638
%K A029635 nonn,tabl,nice,easy
%O A029635 0,3
%A A029635 Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A029635 More terms from David W. Wilson (davidwwilson(AT)comcast.net)

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