A285117 Triangle read by rows: T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) XOR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and XOR is bitwise-XOR (A003987).

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 2, 0, 2, 1, 1, 5, 2, 2, 5, 1, 1, 4, 15, 0, 15, 4, 1, 1, 7, 9, 27, 27, 9, 7, 1, 1, 6, 18, 54, 0, 54, 18, 6, 1, 1, 9, 20, 36, 126, 126, 36, 20, 9, 1, 1, 8, 45, 112, 42, 0, 42, 112, 45, 8, 1, 1, 11, 39, 85, 170, 46, 46, 170, 85, 39, 11, 1, 1, 10, 60, 146, 495, 132, 0, 132, 495, 146, 60, 10, 1

Offset: 0

Author

Antti Karttunen, Apr 16 2017

Keywords

• nonn
• tabl

Examples

			Rows 0 - 12 of the triangle:
  1,
  1, 1,
  1, 0, 1,
  1, 3, 3, 1,
  1, 2, 0, 2, 1,
  1, 5, 2, 2, 5, 1,
  1, 4, 15, 0, 15, 4, 1,
  1, 7, 9, 27, 27, 9, 7, 1,
  1, 6, 18, 54, 0, 54, 18, 6, 1,
  1, 9, 20, 36, 126, 126, 36, 20, 9, 1,
  1, 8, 45, 112, 42, 0, 42, 112, 45, 8, 1,
  1, 11, 39, 85, 170, 46, 46, 170, 85, 39, 11, 1,
  1, 10, 60, 146, 495, 132, 0, 132, 495, 146, 60, 10, 1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10584; the rows 0 - 144 of triangle
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A003987, A007318, A285116, A285118.

Cf. A285114 (row sums).

Programs

Mathematica

T[n_, k_]:= If[n==0 || n==k, 1, BitXor[Binomial[n - 1, k - 1], Binomial[n - 1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 16 2017 *)

PARI

T(n, k) = if (n==0 || n==k, 1, bitxor(binomial(n - 1, k - 1), binomial(n - 1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 16 2017

Scheme

(define (A285117 n) (A285117tr (A003056 n) (A002262 n)))
(define (A285117tr n k) (cond ((zero? k) 1) ((= k n) 1) (else (A003987tr (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A003987bi implements bitwise-XOR (A003987) and A007318tr gives the binomial coefficients (A007318).

Formula

T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) XOR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and XOR is bitwise-XOR (A003987).

T(n,k) = A285116(n,k) - A285118(n,k).

C(n,k) = T(n,k) + 2*A285118(n,k). [Where C(n,k) = A007318.]