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%I #32 Feb 18 2020 12:34:41
%S 1,4,1,15,8,1,56,46,12,1,209,232,93,16,1,780,1091,592,156,20,1,2911,
%T 4912,3366,1200,235,24,1,10864,21468,17784,8010,2120,330,28,1,40545,
%U 91824,89238,48624,16255,3416,441,32,1,151316,386373,430992,275724,111524,29589,5152,568,36,1
%N Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
%C Riordan array (1/(1-4*x+x^2), x/(1-4*x+x^2)).
%C 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.
%C Unsigned version of triangles in A124029 and in A159764.
%C 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
%H Rigoberto Flórez, Leandro Junes, José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, J. Int. Seq. 21 (2018), #18.1.2.
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%F Recurrence: T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
%F Diagonal sums are 4^n = A000302(n).
%F Row sums are A004254(n+1).
%F G.f.: 1/(1-4*x+x^2-y*x)
%F 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).
%F T(n,0) = A001353(n+1).
%e Triangle begins:
%e 1
%e 4, 1
%e 15, 8, 1
%e 56, 46, 12, 1
%e 209, 232, 93, 16, 1
%e 780, 1091, 592, 156, 20, 1
%e 2911, 4912, 3366, 1200, 235, 24, 1
%e 10864, 21468, 17784, 8010, 2120, 330, 28, 1
%e 40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1
%e 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
%e ...
%e Triangle (0, 4, -1/4, 1/4, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
%e 1
%e 0, 1
%e 0, 4, 1
%e 0, 15, 8, 1
%e 0, 56, 46, 12, 1
%e 0, 209, 232, 93, 16, 1
%e ...
%t 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 *)
%Y 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).
%K easy,nonn,tabl
%O 0,2
%A _Philippe Deléham_, Feb 20 2012
%E Offset changed to 0 by _Georg Fischer_, Feb 18 2020
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