A258144 - OEIS

%I #19 Nov 14 2024 08:23:49

%S 1,2,0,-2,5,11,-14,-34,57,127,-209,-461,793,1717,-3002,-6434,11441,

%T 24311,-43757,-92377,167961,352717,-646645,-1352077,2496145,5200301,

%U -9657699,-20058299,37442161,77558761,-145422674,-300540194,565722721,1166803111,-2203961429

%N Alternating row sums of A257241, Stifel's version of the arithmetical triangle.

%H Reinhard Zumkeller, <a href="/A258144/b258144.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{m = 1 .. ceiling(n/2)} (-1)^(m+1)* binomial(n, m), n >= 1.

%F a(2*k+1) = (1 - (-1)^(k+1)*A001791(k)), k >= 0.

%F a(2*k) =  (1 - (-1)^k*A001700(k-1)), k >= 1.

%F O.g.f. for a(2*k+1), k >= 0: (2+3*x - (1-x)*(1+2*x)*c(-x))/((1+4*x)*(1-x)), with the o.g.f. c(x) of A000108 (Catalan).

%F O.g.f. for a(2*(k+1)), k >= 0:

%F   (3+2*x - (1-x)*c(-x))/((1+4*x)*(1-x)).

%F O.g.f. for a(n), n >= 1:

%F x*((1+x)*(2+x+2*x^2) - (1+x+2*x^2)*(1-x^2)*c(-x^2))/((1+4*x^2)*(1-x^2)).

%e n = 3: a(3) = (1 - A001791(1)) = 1 - 1 = 0.

%e n = 4: a(4) = (1 - A001700(1)) = 1 - 3 = -2.

%t Table[Sum[(-1)^(m+1)*Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* _Paolo Xausa_, Nov 14 2024 *)

%o (Haskell)

%o a258144 = sum . zipWith (*) (cycle [1, -1]) . a257241_row

%o -- _Reinhard Zumkeller_, May 22 2015

%Y Cf. A257241, A001700, A001791, A258143, A000108.

%Y Cf. A033999.

%K sign,easy

%O 1,2

%A _Wolfdieter Lang_, May 22 2015