A177693 Triangle, read by rows, T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j).

1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 3, 3, 3, 1, 1, -1, 3, 3, -1, 1, 1, -5, -5, 15, -5, -5, 1, 1, 7, 35, 35, 35, 35, 7, 1, 1, 3, -21, -105, 35, -105, -21, 3, 1, 1, -17, 51, -357, 595, 595, -357, 51, -17, 1, 1, 11, 187, -561, -1309, -6545, -1309, -561, 187, 11, 1

Offset: 0

Author

Roger L. Bagula, May 11 2010

Keywords

• sign
• tabl

Examples

			Triangle begins as:
  1;
  1,   1;
  1,  -1,   1;
  1,  -1,  -1,    1;
  1,   3,   3,    3,     1;
  1,  -1,   3,    3,    -1,     1;
  1,  -5,  -5,   15,    -5,    -5,     1;
  1,   7,  35,   35,    35,    35,     7,    1;
  1,   3, -21, -105,    35,  -105,   -21,    3,   1;
  1, -17,  51, -357,   595,   595,  -357,   51, -17,  1;
  1,  11, 187, -561, -1309, -6545, -1309, -561, 187, 11,  1;
		

References

• Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 47ff.

Links

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

Crossrefs

Cf. A001607.

Programs

Magma

A001607:=[n le 2 select n-1 else -Self(n-1)-2*Self(n-2): n in [1..100]];
p:= func< n | n eq 0 select 1 else (&*[A001607[j+1]: j in [1..n]]) >;
A177693:= func< n,k | p(n)/(p(k)*p(n-k)) >;
[A177693(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

Mathematica

A001607:= LinearRecurrence[{-1,-2}, {0,1}, 100];
p[n_]:= Product[A001607[[i+1]], {i,n}];
T[n_,k_]:= p[n]/(p[k]*p[n-k]);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

A001607=BinaryRecurrenceSequence(-1,-2,0,1)
def p(n): return product(A001607(j) for j in range(1,n+1))
def A177693(n,k): return p(n)/(p(k)*p(n-k))
flatten([[A177693(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

Formula

T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j) and p(0) = 1.

T(n, n-k) = T(n, k).

Extensions

Edited by G. C. Greubel, Apr 08 2024