A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

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, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512

Offset: 0

Author

Roger L. Bagula, Mar 23 2008

Keywords

• easy
• tabl
• sign

Examples

			Triangle begins as:
   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,   16,   32;
  -1, -7,  18,   56,  -48, -112,   32,   64;
   1, -7, -32,   56,  160, -112, -256,   64,   128;
   1,  9, -32, -120,  160,  432, -256, -576,   128, 256;
  -1,  9,  50, -120, -400,  432, 1120, -576, -1280, 256, 512;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.

Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.

Cf. A124182.

Programs

Magma

function A053120(n,k)
  if ((n+k) mod 2) eq 1 then return 0;
  elif n eq 0 then return 1;
  else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
  end if;
end function;
A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
[A136523(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023

Mathematica

A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A053120(n,k):
    if (n+k)%2==1: return 0
    elif n==0: return 1
    else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023

Formula

T(n, k) = A053120(n,k) + A053120(n-1,k).

Sum_{k=0..n} T(n, k) = A040000(n).

From G. C. Greubel, Jul 26 2023: (Start)

T(n, 0) = A057077(n).

T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).

T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).

T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).

T(n, n) = A011782(n).

T(n, n-1) = A011782(n-1).

T(n, n-2) = -A001792(n-2).

T(n, n-4) = A001793(n-3).

T(n, n-6) = -A001794(n-6).

Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)

Extensions

Edited by G. C. Greubel, Jul 26 2023