cp's OEIS Frontend

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

%I A258143 #30 Nov 14 2024 08:22:43
%S A258143 1,2,6,10,25,41,98,162,381,637,1485,2509,5811,9907,22818,39202,89845,
%T A258143 155381,354521,616665,1401291,2449867,5546381,9740685,21977515,
%U A258143 38754731,87167163,154276027,345994215,614429671,1374282018,2448023842,5461770405,9756737701,21717436833
%N A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.
%C A258143 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
%H A258143 Reinhard Zumkeller, <a href="/A258143/b258143.txt">Table of n, a(n) for n = 1..1000</a>
%F A258143 a(n) = Sum_{m = 1 .. ceiling(n/2)} binomial(n, m), n >= 1.
%F A258143 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
%F A258143 a(2*k+1) = 2^(2*k+1) - (1 + A008549(k)), k >= 0.
%F A258143 a(2*k) = 2^(2*k) - (1 + A000346(k-1)), k >= 1.
%F A258143 O.g.f.: x*(2+3*x+x^2 - (1-x^2)*(1+x)*c(x^2))/((1-(2*x)^2)*(1-x^2)) where c(x) is the o.g.f. of A000108.
%F A258143 O.g.f. for a(2*k+1), k >= 0: (2+x - (1-x)*c(x))/ ((1-4*x)*(1-x)).
%F A258143 O.g.f. for a(2*(k+1)), k >= 0: (3 - (1-x)*c(x))/ ((1-4*x)*(1-x)).
%F A258143 a(n) = A116406(n+1) - 1. - _Hugo Pfoertner_, Nov 14 2024
%e A258143 n=3: a(3) = 2^3 - (1 + A008549(1)) = 8 - (1 + 1) = 6.
%e A258143 n=4: a(4) = 2^4 - (1 + A000346(1)) = 16 - (1 +  5) = 10.
%t A258143 Table[Sum[Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* _Paolo Xausa_, Nov 14 2024 *)
%o A258143 (Haskell)
%o A258143 a258143 = sum . a257241_row  -- _Reinhard Zumkeller_, May 22 2015
%Y A258143 Cf. A257241, A000346, A008549, A000108, A116406, A258144.
%K A258143 nonn,easy
%O A258143 1,2
%A A258143 _Wolfdieter Lang_, May 22 2015