Search: id:A034839
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%I A034839
%S A034839 1,1,1,1,1,3,1,6,1,1,10,5,1,15,15,1,1,21,35,7,1,28,70,28,1,1,36,126,
%T A034839 84,9,1,45,210,210,45,1,1,55,330,462,165,11,1,66,495,924,495,66,1,1,
%U A034839 78,715,1716,1287,286,13
%N A034839 Triangular array formed by taking every other term of each row of Pascal's 
               triangle.
%C A034839 Number of compositions of n having k parts greater than 1. Example: T(5,
               2)=5 because we have 3+2, 2+3, 2+2+1, 2+1+2 and 1+2+2. Number of 
               binary words of length n-1 having k runs of consecutive 1's. Example: 
               T(5,2)=5 because we have 1010, 1001, 0101, 1101 and 1011. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2005
%C A034839 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2008: 
               (Start)
%C A034839 Received from Herb Conn; Custer, SD:
%C A034839 Let T = tan x, then
%C A034839 tan x = T
%C A034839 tan 2x = 2T / (1 - T^2)
%C A034839 tan 3x = (3T - T^3) / (1 - 3T^2)
%C A034839 tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
%C A034839 tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
%C A034839 tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
%C A034839 tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
%C A034839 tax 8x = (8T - 56T^3 + 56T^5 - 8T^7) /
%C A034839 (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
%C A034839 tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) /
%C A034839 (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
%C A034839 ... To get the next one in the series, (tan 10x), for the numerator add:
%C A034839 9....84....126....36....1 previous numerator +
%C A034839 1....36....126....84....9 previous denominator =
%C A034839 10..120....252...120...10 = new numerator
%C A034839 For the denominator add:
%C A034839 ......9.....84...126...36...1 = previous numerator +
%C A034839 1....36....126....84....9.... = previous denominator =
%C A034839 1....45....210...210...45...1 = new denominator
%C A034839 ...where numerators = A034867, denominators = A034839
%C A034839 (End)
%H A034839 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Tangent.html">Tangent</a> [From Eric W. Weisstein (eric(AT)weisstein.com), 
               Oct 18 2008]
%F A034839 E.g.f.: exp(x)*cosh(x*sqrt(y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Mar 20 2005
%F A034839 T(n, k)=binomial(n, 2k). G.f.=(1-z)/[(1-z)^2-tz^2]. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Mar 30 2005
%e A034839 1; 1; 1, 1; 1, 3; 1, 6, 1; 1, 10, 5; 1, 15, 15, 1;...
%p A034839 for n from 0 to 13 do seq(binomial(n,2*k),k=0..floor(n/2)) od;# yields 
               sequence in triangular form (Deutsch)
%Y A034839 Cf. A007318.
%Y A034839 Cf. A034867.
%Y A034839 Cf. A034867.
%Y A034839 Sequence in context: A127096 A128489 A130541 this_sequence A089732 A158905 
               A098076
%Y A034839 Adjacent sequences: A034836 A034837 A034838 this_sequence A034840 A034841 
               A034842
%K A034839 nonn,easy,tabf
%O A034839 0,6
%A A034839 N. J. A. Sloane (njas(AT)research.att.com).

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