Search: id:A008315
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%I A008315
%S A008315 1,1,1,1,1,2,1,3,2,1,4,5,1,5,9,5,1,6,14,14,1,7,20,28,14,1,8,27,48,42,
%T A008315 1,9,35,75,90,42,1,10,44,110,165,132,1,11,54,154,275,297,132,1,12,65,
%U A008315 208,429,572,429,1,13,77,273,637,1001,1001,429,1,14,90,350,910,1638,2002,
               1430,1,15,104
%N A008315 Catalan triangle. Also triangle of expansions of powers of x in terms 
               of Chebyshev polynomials U_n (x).
%C A008315 There are several versions of a Catalan triangle: see A009766, A008315, 
               A028364.
%C A008315 Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: 
               T(4,1)=3 because in th top row we can have 124, 134, or 123 (but 
               not 234). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 23 2004
%C A008315 T(n,k) is the number of n-digit binary words (length n sequences on {0,
               1}) containing k 1's such that no initial segment of the sequence 
               has more 1's than 0's. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jul 31 2009]
%D A008315 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 796.
%D A008315 K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc., 10 
               (1997), 139-167.
%D A008315 P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 
               30, Nos. 3/4, 1994, 52-54.
%H A008315 T. D. Noe, <a href="b008315.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A008315 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008315 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</
               a>
%H A008315 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A008315 T(n, 0)=1 if n >= 0; T(2k, k)=T(2k-1, k-1) if k>0; T(n, k)=T(n-1, k-1)+T(n-1, 
               k) if k=1, 2, ...[ n/2 ].
%F A008315 T(n, k) = C(n, k)-C(n, k-1) where C(n, k) is binomial coefficient.
%F A008315 Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 
               = A000108(n); A000108 : Catalan numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 15 2004
%F A008315 T(n,k)=Binomial(n,k)*(n-2k+1)/(n-k+1) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jul 31 2009]
%e A008315 Triangle begins:
%e A008315 1;
%e A008315 1;
%e A008315 1,1;
%e A008315 1,2;
%e A008315 1,3,2;
%e A008315 1,4,5;
%e A008315 1,5,9,5;
%e A008315 1,6,14,14;
%e A008315 1,7,20,28,14;
%e A008315 ...
%e A008315 T(5,2)=5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 
               1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0} [From 
               Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 31 2009]
%t A008315 Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/
               2]}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jul 31 2009]
%o A008315 (PARI) T(n,k)=if(k<0|k>n\2, 0, if(n==0, 1, T(n-1,k-1) + T(n-1,k))).
%Y A008315 T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial 
               coefficients).
%Y A008315 This is also the positive half of the triangle in A008482 - Michael Somos
%Y A008315 This is another reading (by shallow diagonals) of the triangle A009766, 
               i.e. A008315[n] = A009766[A056536[n]].
%Y A008315 Sequence in context: A165999 A049280 A108786 this_sequence A029635 A104741 
               A167237
%Y A008315 Adjacent sequences: A008312 A008313 A008314 this_sequence A008316 A008317 
               A008318
%K A008315 nonn,tabf,nice,easy
%O A008315 0,6
%A A008315 N. J. A. Sloane (njas(AT)research.att.com).
%E A008315 Expanded description from Clark Kimberling Jun 15 1997

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