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%I #15 Jul 27 2023 08:21:26
%S 1,1,1,-1,1,2,-1,-3,2,4,1,-3,-8,4,8,1,5,-8,-20,8,16,-1,5,18,-20,-48,
%T 16,32,-1,-7,18,56,-48,-112,32,64,1,-7,-32,56,160,-112,-256,64,128,1,
%U 9,-32,-120,160,432,-256,-576,128,256,-1,9,50,-120,-400,432,1120,-576,-1280,256,512
%N Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.
%H G. C. Greubel, <a href="/A136523/b136523.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = A053120(n,k) + A053120(n-1,k).
%F Sum_{k=0..n} T(n, k) = A040000(n).
%F From _G. C. Greubel_, Jul 26 2023: (Start)
%F T(n, 0) = A057077(n).
%F T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
%F T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
%F T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
%F T(n, n) = A011782(n).
%F T(n, n-1) = A011782(n-1).
%F T(n, n-2) = -A001792(n-2).
%F T(n, n-4) = A001793(n-3).
%F T(n, n-6) = -A001794(n-6).
%F Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e -1, 1, 2;
%e -1, -3, 2, 4;
%e 1, -3, -8, 4, 8;
%e 1, 5, -8, -20, 8, 16;
%e -1, 5, 18, -20, -48, 16, 32;
%e -1, -7, 18, 56, -48, -112, 32, 64;
%e 1, -7, -32, 56, 160, -112, -256, 64, 128;
%e 1, 9, -32, -120, 160, 432, -256, -576, 128, 256;
%e -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512;
%t A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
%t T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o function A053120(n,k)
%o if ((n+k) mod 2) eq 1 then return 0;
%o elif n eq 0 then return 1;
%o else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
%o end if;
%o end function;
%o A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
%o [A136523(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 26 2023
%o (SageMath)
%o def A053120(n,k):
%o if (n+k)%2==1: return 0
%o elif n==0: return 1
%o else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
%o def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
%o flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 26 2023
%Y Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.
%Y Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.
%Y Cf. A124182.
%K easy,tabl,sign
%O 0,6
%A _Roger L. Bagula_, Mar 23 2008
%E Edited by _G. C. Greubel_, Jul 26 2023
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