A206603 Maximal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}.

0, 0, 1, 4, 13, 36, 94, 232, 557, 1300, 2986, 6744, 15074, 33320, 73116, 159184, 344701, 742068, 1590898, 3395320, 7222550, 15308920, 32362276, 68213424, 143463378, 300999816, 630353764, 1317415792, 2748991012, 5726300880, 11911913912, 24742452128, 51331847709

Offset: 0

Author

Alois P. Heinz, Feb 10 2012

Keywords

• nonn

Comments

The base row of the addition triangle contains a permutation of the n+1 integers or half-integers {k-n/2, k=0..n}. Each number in a higher row is the sum of the two numbers directly below it. Rows above the base row contain only integers. The base row consists of integers iff n is even.

Because of symmetry, a(n) is also the absolute value of the minimal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}.

a(n) is odd iff n = 2^m and m > 0.

Examples

			a(3) =  4:   max:      4            min:    -4
                    1    3               -1   -3
                -1    2    1           1   -2   -1
            -3/2  1/2  3/2 -1/2    3/2 -1/2 -3/2  1/2
a(4) = 13:   max:     13            min:   -13
                     5  8                 -5 -8
                   0  5  3               0 -5 -3
                -2  2  3  0            2 -2 -3  0
              -2  0  2  1 -1         2  0 -2 -1  1
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001787, A189390, A189391, A206604.

Programs

Maple

a:= n-> add (binomial(n, floor(k/2))*(k-n/2), k=0..n):
seq (a(n), n=0..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n*(n-1)/2,
      ((2*n^2-6)*a(n-1) +4*(n-1)*(n-4)*a(n-2)
       -8*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
    end:
seq(a(n), n=0..40); # Alois P. Heinz, Apr 25 2013

Mathematica

a = DifferenceRoot[Function[{y, n}, {8(n+1)(n+2)y[n] - 4(n-1)(n+2)y[n+1] - (2n^2 + 12n + 12)y[n+2] + (n+1)(n+3)y[n+3] == 0, y[0] == 0, y[1] == 0, y[2] == 1, y[3] == 4}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

PARI

a(n) = sum(k=0, n, binomial(n, k\2)*(k-n/2)); \\ Michel Marcus, Dec 20 2020

Python

from math import comb
def A206603(n): return sum(comb(n,k>>1)*((k<<1)-n) for k in range(n+1))>>1 # Chai Wah Wu, Oct 28 2024

Formula

a(n) = Sum_{k=0..n} C(n,floor(k/2)) * (k-n/2).

G.f.: (1-sqrt(1-4*x^2)) / (2*(2*x-1)^2).

a(n) = A189390(n)-A001787(n) = A001787(n)-A189391(n) = (A189390(n)-A189391(n))/2 = (A206604(n)-1)/2.