|
%I #28 Oct 01 2022 23:23:25
%S 1,1,0,1,1,0,1,1,0,0,1,2,1,0,0,1,2,1,0,0,0,1,3,3,1,0,0,0,1,3,3,1,0,0,
%T 0,0,1,4,6,4,1,0,0,0,0,1,4,6,4,1,0,0,0,0,0,1,5,10,10,5,1,0,0,0,0,0,1,
%U 5,10,10,5,1,0,0,0,0,0,0,1,6,15,20,15,6,1,0,0,0,0,0,0,1,6,15,20,15,6,1,0,0,0
%N Triangle T(n,k), read by rows given by [1,0,-1,0,0,0,0,0,0,...] DELTA [0,1,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Triangle read by rows, Pascal's triangle (A007318) rows repeated.
%C Riordan array (1/(1-x), x^2/(1-x^2)). - _Philippe Deléham_, Feb 27 2012
%H Reinhard Zumkeller, <a href="/A152815/b152815.txt">Rows n = 0..150 of triangle, flattened</a>
%F T(n,k) = T(n-1,k) + ((1+(-1)^n)/2)*T(n-1,k-1).
%F G.f.: (1+x)/(1-(1+y)*x^2).
%F Sum_{k=0..n} T(n,k)*x^k = A000012(n), A016116(n), A108411(n), A213173(n), A074872(n+1) for x = 0,1,2,3,4 respectively. - _Philippe Deléham_, Nov 26 2011, Apr 22 2013
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 1, 0, 0;
%e 1, 2, 1, 0, 0;
%e 1, 2, 1, 0, 0, 0;
%e 1, 3, 3, 1, 0, 0, 0;
%e 1, 3, 3, 1, 0, 0, 0, 0;
%e 1, 4, 6, 4, 1, 0, 0, 0, 0; ...
%t m = 13;
%t (* DELTA is defined in A084938 *)
%t DELTA[Join[{1, 0, -1}, Table[0, {m}]], Join[{0, 1, -1}, Table[0, {m}]], m] // Flatten (* _Jean-François Alcover_, Feb 19 2020 *)
%t T[n_, k_] := If[n<0, 0, Binomial[Floor[n/2], k]]; (* _Michael Somos_, Oct 01 2022 *)
%o (Haskell)
%o a152815 n k = a152815_tabl !! n !! k
%o a152815_row n = a152815_tabl !! n
%o a152815_tabl = [1] : [1,0] : t [1,0] where
%o t ys = zs : zs' : t zs' where
%o zs' = zs ++ [0]; zs = zipWith (+) ([0] ++ ys) (ys ++ [0])
%o -- _Reinhard Zumkeller_, Feb 28 2012
%o {T(n, k) = if(n<0, 0, binomial(n\2, k))}; /* _Michael Somos_, Oct 01 2022 */
%Y Cf. A007318, A064861, A152198 (another version), A000931 (diagonal sums), A016116 (row sums).
%K easy,nonn,tabl
%O 0,12
%A _Philippe Deléham_, Dec 13 2008
%E Example corrected by _Philippe Deléham_, Dec 13 2008
|
