A285117 - OEIS

%I #14 Apr 17 2017 09:03:11

%S 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,

%T 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,

%U 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

%N 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).

%H Antti Karttunen, <a href="/A285117/b285117.txt">Table of n, a(n) for n = 0..10584; the rows 0 - 144 of triangle</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F 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).

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

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

%e Rows 0 - 12 of the triangle:

%e   1,

%e   1, 1,

%e   1, 0, 1,

%e   1, 3, 3, 1,

%e   1, 2, 0, 2, 1,

%e   1, 5, 2, 2, 5, 1,

%e   1, 4, 15, 0, 15, 4, 1,

%e   1, 7, 9, 27, 27, 9, 7, 1,

%e   1, 6, 18, 54, 0, 54, 18, 6, 1,

%e   1, 9, 20, 36, 126, 126, 36, 20, 9, 1,

%e   1, 8, 45, 112, 42, 0, 42, 112, 45, 8, 1,

%e   1, 11, 39, 85, 170, 46, 46, 170, 85, 39, 11, 1,

%e   1, 10, 60, 146, 495, 132, 0, 132, 495, 146, 60, 10, 1

%t 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 *)

%o (Scheme)

%o (define (A285117 n) (A285117tr (A003056 n) (A002262 n)))

%o (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).

%o (PARI) T(n, k) = if (n==0 || n==k, 1, bitxor(binomial(n - 1, k - 1), binomial(n - 1, k)));

%o for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ _Indranil Ghosh_, Apr 16 2017

%Y Cf. A003987, A007318, A285116, A285118.

%Y Cf. A285114 (row sums).

%K nonn,tabl

%O 0,8

%A _Antti Karttunen_, Apr 16 2017