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A207823
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Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
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8
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1, 4, 1, 15, 8, 1, 56, 46, 12, 1, 209, 232, 93, 16, 1, 780, 1091, 592, 156, 20, 1, 2911, 4912, 3366, 1200, 235, 24, 1, 10864, 21468, 17784, 8010, 2120, 330, 28, 1, 40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1, 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
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OFFSET
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0,2
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COMMENTS
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Riordan array (1/(1-4*x+x^2), x/(1-4*x+x^2)).
Subtriangle of the triangle given by (0, 4, -1/4, 1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Unsigned version of triangles in A124029 and in A159764.
For 1<=k<=n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3,4} containing k-1 letters equal 4 and avoiding 01. - Milan Janjic, Dec 20 2016
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LINKS
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Table of n, a(n) for n=0..54.
Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, J. Int. Seq. 21 (2018), #18.1.2.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
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FORMULA
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Recurrence: T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
Diagonal sums are 4^n = A000302(n).
Row sums are A004254(n+1).
G.f.: 1/(1-4*x+x^2-y*x)
T(n,n) = 1, T(n+1,n) = 4*n+4 = A008586(n+1), T(n+2,n) = (n+1)*(8n+15) = A139278(n+1).
T(n,0) = A001353(n+1).
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EXAMPLE
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Triangle begins:
1
4, 1
15, 8, 1
56, 46, 12, 1
209, 232, 93, 16, 1
780, 1091, 592, 156, 20, 1
2911, 4912, 3366, 1200, 235, 24, 1
10864, 21468, 17784, 8010, 2120, 330, 28, 1
40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1
151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
...
Triangle (0, 4, -1/4, 1/4, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
1
0, 1
0, 4, 1
0, 15, 8, 1
0, 56, 46, 12, 1
0, 209, 232, 93, 16, 1
...
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MATHEMATICA
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With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 4 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
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CROSSREFS
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Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).
Sequence in context: A095307 A159764 A124029 * A056920 A123382 A197653
Adjacent sequences: A207820 A207821 A207822 * A207824 A207825 A207826
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Philippe Deléham, Feb 20 2012
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EXTENSIONS
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Offset changed to 0 by Georg Fischer, Feb 18 2020
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STATUS
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approved
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