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 A034839 Triangular array formed by taking every other term of each row of Pascal's triangle. 36
 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, 84, 9, 1, 45, 210, 210, 45, 1, 1, 55, 330, 462, 165, 11, 1, 66, 495, 924, 495, 66, 1, 1, 78, 715, 1716, 1287, 286, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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, Mar 30 2005 From Gary W. Adamson, Oct 17 2008: (Start) Received from Herb Conn: 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) tan 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) Triangle, with zeros omitted, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011 The row (1,66,495,924,495,66,1) plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534. - Tom Copeland, Dec 12 2016 Binomial(n,2k) is also the number of permutations avoiding both 123 and 132 with k ascents, i.e., positions with w[i]= 0 and k = 0, 1, ... floor(n/2). G.f.: (1-z)/((1-z)^2 - t*z^2). (End) O.g.f. for column no. k is (1/(1-x))*(x/(1-x))^(2*k), k >= 0 [from the g.f. given in the preceding formula]. - Wolfdieter Lang, Jan 18 2013 From Peter Bala, Jul 14 2015: (Start) Stretched Riordan array ( 1/(1 - x ), x^2/(1 - x)^2 ) in the terminology of Corsani et al. Denote this array by P. Then P * A007318 = A201701. P * transpose(P) is A119326 read as a square array. Let Q denote the array ( (-1)^k*binomial(2*n,k) )n,k>=0. Q is a signed version of A034870. Then Q*P = the identity matrix, that is, Q is a left-inverse array of P (see Corsani et al., p. 111). P * A034870 = A080928. (End) Even rows are A086645. An aerated version of this array is A099174 with each diagonal divided by the first element of the diagonal, the double factorials A001147. - Tom Copeland, Dec 12 2015 EXAMPLE Triangluar array T(n, k) begins:   1   1   1  1   1  3   1  6  1   1 10  5   1 15 15 1 ... - Philippe Deléham, Dec 12 2011 MAPLE for n from 0 to 13 do seq(binomial(n, 2*k), k=0..floor(n/2)) od; # yields sequence in triangular form; # Emeric Deutsch, Mar 30 2005 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 12; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x] v[n_, x_] := u[n - 1, x] + v[n - 1, x] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu]  (* A034839 as a triangle *) cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv]  (* A034867 as a triangle *) (* Clark Kimberling, Feb 18 2012 *) Table[Binomial[n, k], {n, 0, 13}, {k, 0, Floor[n, 2], 2}] // Flatten (* Michael De Vlieger, Dec 13 2016 *) PROG (PARI) for(n=0, 15, for(k=0, floor(n/2), print1(binomial(n, 2*k), ", "))) \\ G. C. Greubel, Feb 23 2018 (MAGMA) /* As a triangle */ [[Binomial(n, 2*k):k in [0..Floor(n/2)]] : n in [0..10]]; // G. C. Greubel, Feb 23 2018 CROSSREFS Cf. A007318, A034867, A034870, A080928, A119326, A201701. Cf. A008619 (row lengths), A086645. Sequence in context: A127096 A128489 A130541 * A089732 A158905 A098076 Adjacent sequences:  A034836 A034837 A034838 * A034840 A034841 A034842 KEYWORD nonn,easy,tabf AUTHOR STATUS approved

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Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)