A285116 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) OR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and OR is bitwise-OR (A003986).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 3, 1, 1, 5, 6, 6, 5, 1, 1, 5, 15, 10, 15, 5, 1, 1, 7, 15, 31, 31, 15, 7, 1, 1, 7, 23, 55, 35, 55, 23, 7, 1, 1, 9, 28, 60, 126, 126, 60, 28, 9, 1, 1, 9, 45, 116, 126, 126, 126, 116, 45, 9, 1, 1, 11, 47, 125, 250, 254, 254, 250, 125, 47, 11, 1, 1, 11, 63, 183, 495, 462, 462, 462, 495, 183, 63, 11, 1

Offset: 0

Author

Antti Karttunen, Apr 16 2017

Keywords

• nonn
• tabl

Examples

			Rows 0 - 12 of the triangle:
  1,
  1, 1,
  1, 1, 1,
  1, 3, 3, 1,
  1, 3, 3, 3, 1,
  1, 5, 6, 6, 5, 1,
  1, 5, 15, 10, 15, 5, 1,
  1, 7, 15, 31, 31, 15, 7, 1,
  1, 7, 23, 55, 35, 55, 23, 7, 1,
  1, 9, 28, 60, 126, 126, 60, 28, 9, 1,
  1, 9, 45, 116, 126, 126, 126, 116, 45, 9, 1,
  1, 11, 47, 125, 250, 254, 254, 250, 125, 47, 11, 1,
  1, 11, 63, 183, 495, 462, 462, 462, 495, 183, 63, 11, 1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10439; the rows 0 - 143 of triangle
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A003986, A007318, A285117, A285118.

Cf. A285113 (row sums).

Programs

Mathematica

T[n_, k_]:= If[n==0 || n==k, 1, BitOr[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 *)

PARI

T(n, k) = if (n==0 || n==k, 1, bitor(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

Scheme

(define (A285116 n) (A285116tr (A003056 n) (A002262 n)))
(define (A285116tr n k) (cond ((zero? k) 1) ((= k n) 1) (else (A003986tr (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A003986bi implements bitwise-OR (A003986) and A007318tr gives the binomial coefficients (A007318).

Formula

T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) OR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and OR is bitwise-OR (A003986).

T(n,k) = A285117(n,k) + A285118(n,k).

C(n,k) = T(n,k) + A285118(n,k). [Where C(n,k) = A007318.]