A285118 - OEIS

%I #12 Apr 17 2017 09:03:18

%S 0,0,0,0,1,0,0,0,0,0,0,1,3,1,0,0,0,4,4,0,0,0,1,0,10,0,1,0,0,0,6,4,4,6,

%T 0,0,0,1,5,1,35,1,5,1,0,0,0,8,24,0,0,24,8,0,0,0,1,0,4,84,126,84,4,0,1,

%U 0,0,0,8,40,80,208,208,80,40,8,0,0,0,1,3,37,0,330,462,330,0,37,3,1,0,0,0,0,64,204,264,792,792,264,204,64,0,0,0

%N Triangle read by rows: T(0,n) = T(n,n) = 0; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) AND C(n-1,k), where C(n,k) is binomial coefficient (A007318) & AND is bitwise-AND (A004198).

%H Antti Karttunen, <a href="/A285118/b285118.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) = 0; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) AND C(n-1,k), where C(n,k) is binomial coefficient (A007318) & AND is bitwise-AND (A004198).

%F T(n,k) = A285116(n,k) - A285117(n,k).

%F A007318(n,k) = C(n,k) = A285116(n,k) + T(n,k) = A285117(n,k) + 2*T(n,k).

%e Rows 0-13 of array:

%e   0,

%e   0, 0,

%e   0, 1, 0,

%e   0, 0, 0, 0,

%e   0, 1, 3, 1, 0,

%e   0, 0, 4, 4, 0, 0,

%e   0, 1, 0, 10, 0, 1, 0,

%e   0, 0, 6, 4, 4, 6, 0, 0,

%e   0, 1, 5, 1, 35, 1, 5, 1, 0,

%e   0, 0, 8, 24, 0, 0, 24, 8, 0, 0,

%e   0, 1, 0, 4, 84, 126, 84, 4, 0, 1, 0,

%e   0, 0, 8, 40, 80, 208, 208, 80, 40, 8, 0, 0,

%e   0, 1, 3, 37, 0, 330, 462, 330, 0, 37, 3, 1, 0,

%e   0, 0, 0, 64, 204, 264, 792, 792, 264, 204, 64, 0, 0, 0

%t T[n_, k_]:= If[n==0 || n==k, 0, BitAnd[Binomial[n - 1, k - 1], Binomial[n - 1, k]]]; Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Indranil Ghosh_, Apr 16 2017 *)

%o (Scheme)

%o (define (A285118 n) (A285118tr (A003056 n) (A002262 n)))

%o (define (A285118tr n k) (cond ((zero? k) 0) ((= k n) 0) (else (A004198bi (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A004198bi implements bitwise-AND (A004198) and A007318tr gives the binomial coefficients (A007318).

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

%o for(n=0, 13, for(k=0, n, print1(T(n, k),", ");); print();) \\ _Indranil Ghosh_, Apr 16 2017

%Y Cf. A004198, A007318, A285116, A285117.

%Y Cf. A285115 (row sums).

%K nonn,tabl

%O 0,13

%A _Antti Karttunen_, Apr 16 2017