A258144 Alternating row sums of A257241, Stifel's version of the arithmetical triangle.
1, 2, 0, -2, 5, 11, -14, -34, 57, 127, -209, -461, 793, 1717, -3002, -6434, 11441, 24311, -43757, -92377, 167961, 352717, -646645, -1352077, 2496145, 5200301, -9657699, -20058299, 37442161, 77558761, -145422674, -300540194, 565722721, 1166803111, -2203961429
Offset: 1
Examples
n = 3: a(3) = (1 - A001791(1)) = 1 - 1 = 0. n = 4: a(4) = (1 - A001700(1)) = 1 - 3 = -2.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Programs
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Haskell
a258144 = sum . zipWith (*) (cycle [1, -1]) . a257241_row -- Reinhard Zumkeller, May 22 2015
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Mathematica
Table[Sum[(-1)^(m+1)*Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* Paolo Xausa, Nov 14 2024 *)
Formula
a(n) = Sum_{m = 1 .. ceiling(n/2)} (-1)^(m+1)* binomial(n, m), n >= 1.
a(2*k+1) = (1 - (-1)^(k+1)*A001791(k)), k >= 0.
a(2*k) = (1 - (-1)^k*A001700(k-1)), k >= 1.
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).
O.g.f. for a(2*(k+1)), k >= 0:
(3+2*x - (1-x)*c(-x))/((1+4*x)*(1-x)).
O.g.f. for a(n), n >= 1:
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)).