0,6

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

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2008: (Start)

Received from Herb Conn; Custer, SD:

Let T = tan x, then

tan x = T

tan 2x = 2T / (1 - T^2)

tan 3x = (3T - T^3) / (1 - 3T^2)

tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)

tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)

tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)

tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)

tax 8x = (8T - 56T^3 + 56T^5 - 8T^7) /

(1 - 28T^2 + 70T^4 - 28T^6 + T^8)

tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) /

(1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)

... To get the next one in the series, (tan 10x), for the numerator add:

9....84....126....36....1 previous numerator +

1....36....126....84....9 previous denominator =

10..120....252...120...10 = new numerator

For the denominator add:

......9.....84...126...36...1 = previous numerator +

1....36....126....84....9.... = previous denominator =

1....45....210...210...45...1 = new denominator

...where numerators = A034867, denominators = A034839

(End)

Eric Weisstein's World of Mathematics, Tangent [From Eric W. Weisstein (eric(AT)weisstein.com), Oct 18 2008]

E.g.f.: exp(x)*cosh(x*sqrt(y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 20 2005

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

1; 1; 1, 1; 1, 3; 1, 6, 1; 1, 10, 5; 1, 15, 15, 1;...

for n from 0 to 13 do seq(binomial(n, 2*k), k=0..floor(n/2)) od; # yields sequence in triangular form (Deutsch)

Cf. A007318.

Cf. A034867.

Cf. A034867.

Sequence in context: A127096 A128489 A130541 this_sequence A089732 A158905 A098076

Adjacent sequences: A034836 A034837 A034838 this_sequence A034840 A034841 A034842

nonn,easy,tabf

N. J. A. Sloane (njas(AT)research.att.com).