%I #21 Apr 26 2018 10:07:08
%S 1,5,1,24,10,1,115,73,15,1,551,470,147,20,1,2640,2828,1190,246,25,1,
%T 12649,16310,8631,2400,370,30,1,60605,91371,58275,20385,4225,519,35,1,
%U 290376,501150,374115,157800,41140,6790,693,40,1
%N Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
%C Riordan array (1/(1-5*x+x^2), x/(1-5*x+x^2)).
%C Subtriangle of triangle given by (0, 5, -1/5, 1/5, 0, 0, 0, 0, 0, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C Unsigned version of A123967 and A179900.
%C For 1<=k<=n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3,4,5} containing k-1 letters equal 5 and avoiding 01. - _Milan Janjic_, Dec 20 2016
%H Michael De Vlieger, <a href="/A207824/b207824.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150)
%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) = 5*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
%F G.f.: 1/(1-5*x+x^2-y*x).
%F Diagonal sums are 5^n = A000351(n).
%F Row sums are A001109(n+1).
%F T(n,0) = A004254(n+1), T(n+1,n) = 5n+5 = A008587(n+1).
%e Triangle begins :
%e 1
%e 5, 1
%e 24, 10, 1
%e 115, 73, 15, 1
%e 551, 470, 147, 20, 1
%e 2640, 2828, 1190, 246, 25, 1
%e 12649, 16310, 8631, 2400, 370, 30, 1
%e ...
%e Triangle (0, 5, -1/5, 1/5, 0, 0, 0,...) DELTA (1, 0, 0, 0, ...) begins :
%e 1
%e 0, 1
%e 0, 5, 1
%e 0, 24, 10, 1
%e 0, 115, 73, 15, 1
%e 0, 551, 470, 147, 20, 1
%e 0, 2640, 2828, 1190, 246, 25, 1
%e ...
%t With[{n = 8}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 5 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* _Michael De Vlieger_, Apr 25 2018 *)
%o (PARI) row(n) = Vecrev(polchebyshev(n, 2, (x+5)/2)); \\ _Michel Marcus_, Apr 26 2018
%Y Cf. Triangles 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