A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.

1, 2, 6, 10, 25, 41, 98, 162, 381, 637, 1485, 2509, 5811, 9907, 22818, 39202, 89845, 155381, 354521, 616665, 1401291, 2449867, 5546381, 9740685, 21977515, 38754731, 87167163, 154276027, 345994215, 614429671, 1374282018, 2448023842

Offset

1,2

Comments

a(n) is the number of nonempty subsets of {1,2,...,n} that contain either more odd than even numbers or the same number of odd and even numbers. For example, for n=4, a(4)=10 and the 10 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}; {1,2}, {1,4}, {2,3}, {3,4}, {1,2,3,4}. - Enrique Navarrete, Dec 16 2019

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{m = 1 .. ceiling(n/2)} binomial(n, m), n >= 1.

a(n) = 2^n - 2 - Sum_{i=1..floor(n/2)-1} binomial(n, i), n >= 2; a(1)=1. - Enrique Navarrete, Dec 16 2019

a(2*k+1) = 2^(2*k+1) - (1 + A008549(k)), k >= 0.

a(2*k) = 2^(2*k) - (1 + A000346(k-1)), k >= 1.

O.g.f. for a(2*k+1), k >= 0: (2+x - (1-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 - (1-x)*c(x))/ ((1-4*x)*(1-x)).

O.g.f. for a(n), n >= 1: x*(2+3*x+x^2 - (1-x^2)*(1+x)*c(x^2))/((1-(2*x)^2)*(1-x^2).

Example

n=3: a(3) = 2^3 - (1 + A008549(1)) = 8 - (1 + 1) = 6.
n=4: a(4) = 2^4 - (1 + A000346(1)) = 16 - (1 +  5) = 10.

Prog

(Haskell)
a258143 = sum . a257241_row  -- Reinhard Zumkeller, May 22 2015

Crossrefs

Cf. A257241, A000346, A008549, A000108, A258144.

Sequence in context: A200220 A188224 A104142 * A369290 A079713 A055237

Adjacent sequences: A258140 A258141 A258142 * A258144 A258145 A258146

Keyword

nonn,easy

Author

Wolfdieter Lang, May 22 2015

Status

approved