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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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 A107333

Adjacent sequences:  A285114 A285115 A285116 * A285118 A285119 A285120

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Apr 16 2017

STATUS

approved

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Last modified December 9 08:17 EST 2021. Contains 349627 sequences. (Running on oeis4.)