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A034869
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Right half of Pascal's triangle.
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8
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1, 1, 2, 1, 3, 1, 6, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 21, 7, 1, 70, 56, 28, 8, 1, 126, 84, 36, 9, 1, 252, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 924, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1
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history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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From R. J. Mathar, May 13 2006: (Start)
Also flattened table of the expansion coefficients of x^n in Chebyshev Polynomials T_k(x) of the first kind:
x^n is 2^(1-n) multiplied by the sum of floor(1+n/2) terms using only terms T_k(x) with even k if n even, only terms T_k(x) with odd k if n is odd and halving the coefficient a(..) in front of any T_0(x):
x^0=2^(1-0) a(0)/2 T_0(x)
x^1=2^(1-1) a(1) T_1(x)
x^2=2^(1-2) [a(2)/2 T_0(x)+a(3) T_2(x)]
x^3=2^(1-3) [a(4) T_1(x)+a(5) T_3(x)]
x^4=2^(1-4) [a(6)/2 T_0(x)+a(7) T_2(x) +a(8) T_4(x)]
x^5=2^(1-5) [a(9) T_1(x)+a(10) T_3(x) +a(11) T_5(x)]
x^6=2^(1-6) [a(12)/2 T_0(x)+a(13) T_2(x) +a(14) T_4(x) +a(15) T_6(x)]
x^7=2^(1-7) [a(16) T_1(x)+a(17) T_3(x) +a(18) T_5(x) +a(19) T_7(x)]" (End)
T(n,k) = A034868(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012
Rows are binomial(r-1,(2r+1-(-1)^r)\4 -n ) where r is the row and n is the term. Columns are binomial(2m+c-3,m-1) where c is the column and m is the term. - Anthony Browne, May 17 2016
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LINKS
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Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
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EXAMPLE
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The table starts:
1
1
2 1
3 1
6 4 1
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MAPLE
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for n from 0 to 60 do for j from n mod 2 to n by 2 do print( binomial(n, (n-j)/2) ); od; od; # R. J. Mathar, May 13 2006
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MATHEMATICA
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Table[Binomial[n, k], {n, 0, 14}, {k, Ceiling[n/2], n}] // Flatten (* Michael De Vlieger, May 19 2016 *)
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PROG
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(Haskell)
a034869 n k = a034869_tabf !! n !! k
a034869_row n = a034869_tabf !! n
a034869_tabf = [1] : f 0 [1] where
f 0 us'@(_:us) = ys : f 1 ys where
ys = zipWith (+) us' (us ++ [0])
f 1 vs@(v:_) = ys : f 0 ys where
ys = zipWith (+) (vs ++ [0]) ([v] ++ vs)
Reinhard Zumkeller, improved Dec 21 2015, Jul 27 2012
(PARI) for(n=0, 14, for(k=ceil(n/2), n, print1(binomial(n, k), ", "); ); print(); ) \\ Indranil Ghosh, Mar 31 2017
(Python)
import math
from sympy import binomial
for n in range(15):
print([binomial(n, k) for k in range(math.ceil(n/2), n + 1)]) # Indranil Ghosh, Mar 31 2017
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CROSSREFS
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Cf. A007318, A008619 (row lengths).
Cf. A110654.
Cf. A034868 (left half), A014413, A014462.
Sequence in context: A006241 A336105 A282601 * A205858 A340793 A053222
Adjacent sequences: A034866 A034867 A034868 * A034870 A034871 A034872
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Keyword fixed and example added by Franklin T. Adams-Watters, May 27 2010
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STATUS
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approved
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