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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). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A185664

Adjacent sequences:  A285115 A285116 A285117 * A285119 A285120 A285121

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Apr 16 2017

STATUS

approved

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Last modified July 7 14:34 EDT 2020. Contains 335495 sequences. (Running on oeis4.)