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A034867
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Triangle of odd-numbered terms in rows of Pascal's triangle.
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10
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1, 2, 3, 1, 4, 4, 5, 10, 1, 6, 20, 6, 7, 35, 21, 1, 8, 56, 56, 8, 9, 84, 126, 36, 1, 10, 120, 252, 120, 10, 11, 165, 462, 330, 55, 1, 12, 220, 792, 792, 220, 12, 13, 286, 1287, 1716, 715, 78, 1, 14, 364, 2002, 3432, 2002, 364, 14, 15, 455, 3003, 6435, 5005, 1365, 105, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also triangle of numbers of n-sequences of 0,1 with k subsequences of consecutive 01 because this number is C(n+1,2*k+1). - Roger Cuculiere (cuculier(AT)imaginet.fr), Nov 16 2002
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)
Column k is the sum of columns 2k and 2k+1 of A007318. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]
Triangle, with zeros omitted, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Dec 12 2011
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 136.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
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LINKS
| Eric Weisstein's World of Mathematics, Tangent [From Eric W. Weisstein (eric(AT)weisstein.com), Oct 18 2008]
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FORMULA
| T(n, k) = C(n+1, 2k+1) = Sum[i=k..n-k, C(i, k)*C(n-i, k) ].
E.g.f.: 1+(exp(x)*sinh(x*sqrt(y)))/sqrt(y). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 20 2005
G.f.=1/[(1-z)^2-tz^2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
T(n, k) = Sum_{ j = 0, . . ., n} A034839(j, k). - Philippe DELEHAM, May 18 2005
Pell(n+1) = A000129(n+1) = sum(k=0,...,n) T(n,k) * 2^k = (1/n!)sum(k=0,...,n) A131980(n,k) * 2^k . - Tom Copeland (tcjpn(AT)msn.com), Nov 30 2007
T(n,k)=A007318(n,2k)+A007318(n,2k+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]
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EXAMPLE
| Triangle starts:
1;
2;
3,1;
4,4;
5,10,1;
6,20,6; - DELEHAM Philippe, Dec 12 2011
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MAPLE
| seq(seq(binomial(n+1, 2*k+1), k=0..floor(n/2)), n=0..14); (Deutsch)
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CROSSREFS
| Cf. A007318, A034839.
Sequence in context: A201927 A090244 A096180 * A193790 A055446 A104706
Adjacent sequences: A034864 A034865 A034866 * A034868 A034869 A034870
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
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