A285118 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).

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, 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, 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

Offset

0,13

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) = 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).

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

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

Example

Rows 0-13 of array:
  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, 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, 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

Mathematica

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 *)

Prog

(Scheme)
(define (A285118 n) (A285118tr (A003056 n) (A002262 n)))
(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).
(PARI) T(n, k) = if (n==0 || n==k, 0, bitand(binomial(n - 1, k - 1), binomial(n - 1, k)));
for(n=0, 13, for(k=0, n, print1(T(n, k), ", "); ); print(); ) \\ Indranil Ghosh, Apr 16 2017

Crossrefs

Cf. A004198, A007318, A285116, A285117.

Cf. A285115 (row sums).

Sequence in context: A316836 A058612 A099725 * A128208 A154721 A350449

Adjacent sequences: A285115 A285116 A285117 * A285119 A285120 A285121

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 16 2017

Status

approved