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A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

%I A125662 #43 Mar 30 2024 23:09:26
%S A125662 1,3,1,8,6,1,21,25,9,1,55,90,51,12,1,144,300,234,86,15,1,377,954,951,
%T A125662 480,130,18,1,987,2939,3573,2305,855,183,21,1,2584,8850,12707,10008,
%U A125662 4740,1386,245,24,1,6765,26195,43398,40426,23373,8715,2100,316,27,1
%N A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
%C A125662 Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965.
%C A125662 From _Philippe Deléham_, Feb 19 2012: (Start)
%C A125662 Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)).
%C A125662 Equals A078812*A007318 as infinite lower triangular matrices.
%C A125662 Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). (End)
%C A125662 For 1 <= k <= n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - _Milan Janjic_, Dec 20 2016
%H A125662 Michael De Vlieger, <a href="/A125662/b125662.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150)
%H A125662 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 A125662 T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k); T(0,0)=1; T(n,k)=0 if k < 0 or k > n.
%F A125662 Sum_{k=0..n} T(n, k) = A001353(n+1).
%F A125662 Sum_{k=0..floor(n/2)} T(n-k, k) = A000244(n+1).
%F A125662 G.f.: 1/(1-3*x+x^2-y*x). - _Philippe Deléham_, Feb 19 2012
%F A125662 From _G. C. Greubel_, Aug 20 2023: (Start)
%F A125662 T(n, k) = abs( [x^k]( ChebyshevU(n, (3-x)/2) ) ).
%F A125662 Sum_{k=0..n} (-1)^k*T(n, k) = A000027(n+1).
%F A125662 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000225(n). (End)
%e A125662 Triangle begins:
%e A125662    1;
%e A125662    3,  1;
%e A125662    8,  6,  1;
%e A125662   21, 25,  9,  1;
%e A125662   55, 90, 51, 12,  1;
%e A125662   ...
%e A125662 Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins:
%e A125662   1;
%e A125662   0,  1;
%e A125662   0,  3,  1;
%e A125662   0,  8,  6,  1;
%e A125662   0, 21, 25,  9,  1;
%e A125662   0, 55, 90, 51, 12,  1;
%e A125662   ...
%t A125662 With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* _Michael De Vlieger_, Apr 25 2018 *)
%t A125662 Table[Abs[CoefficientList[ChebyshevU[n,(x-3)/2], x]], {n,0,12}]//Flatten (* _G. C. Greubel_, Aug 20 2023 *)
%o A125662 (Magma)
%o A125662 m:=12;
%o A125662 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A125662 A125662:= func< n,k | Abs( Coefficient(R!( Evaluate(ChebyshevU(n+1), (3-x)/2) ), k) ) >;
%o A125662 [A125662(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Aug 20 2023
%o A125662 (SageMath)
%o A125662 def A125662(n,k): return abs( ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k] )
%o A125662 flatten([[A125662(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 20 2023
%Y A125662 Diagonal sums: A000244(powers of 3).
%Y A125662 Row sums: A001353 (n+1).
%Y A125662 Diagonals: A001906(n+1), A001871.
%Y A125662 Cf. A000012, A008585, A062728.
%Y A125662 Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.
%K A125662 easy,nonn,tabl
%O A125662 0,2
%A A125662 _Philippe Deléham_, Jan 28 2007
%E A125662 a(45) corrected and a(51) added by _Philippe Deléham_, Feb 19 2012