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

0,8

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

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.]

Example

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

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 *)

Prog

(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).
(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

Crossrefs

Cf. A003987, A007318, A285116, A285118.

Cf. A285114 (row sums).

Sequence in context: A319861 A114266 A230206 * A135910 A255916 A367825

Adjacent sequences: A285114 A285115 A285116 * A285118 A285119 A285120

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 16 2017

Status

approved