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A206603 Maximal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

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

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.

EXAMPLE

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

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

CROSSREFS

Cf. A001787, A189390, A189391, A206604.

Sequence in context: A036643 A000299 A102301 * A271176 A031506 A251701

Adjacent sequences:  A206600 A206601 A206602 * A206604 A206605 A206606

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 10 2012

STATUS

approved

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Last modified December 14 01:03 EST 2019. Contains 329977 sequences. (Running on oeis4.)