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A189390 The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it. 8
0, 1, 5, 16, 45, 116, 286, 680, 1581, 3604, 8106, 18008, 39650, 86568, 187804, 404944, 868989, 1856180, 3950194, 8376056, 17708310, 37329016, 78499620, 164682416, 344789970, 720430216, 1502768996, 3129355120, 6507087396, 13510929104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The maximal value is reached when the largest numbers are placed in the middle and the smallest numbers at the border of the first row, i.e., [0,2,...,n,...,3,1]. Since the value of the apex is given as sum(c_k binomial(n,k)), one can compute this maximal value directly.

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..1000

Steven Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.

FORMULA

a(n) = Sum_{k=0..floor((n-1)/2)} (4*k+1)*C(n,k) + (n+1 mod 2)*n*C(n,n/2).

a(n) = n*2^n-A189391(n). - M. F. Hasler, Jan 24 2012

a(n) = Sum_{k=0..n} k * C(n,floor(k/2)) = Sum_{k=0..n} k*A107430(n,k). - Alois P. Heinz, Feb 02 2012

G.f.: (2*x-sqrt(1-4*x^2)+1) / (2*(2*x-1)^2). - Alois P. Heinz, Feb 09 2012

D-finite with recurrence n*a(n) -4*n*a(n-1) +12*a(n-2) +16*(n-3)*a(n-3) +16*(-n+3)*a(n-4)=0. - R. J. Mathar, Jul 28 2016

D-finite with recurrence n*(2*n-3)*a(n) +2*(-2*n^2-n+5)*a(n-1) +4*(-2*n^2+9*n-5)*a(n-2) +8*(2*n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 28 2016

a(n) = Sum_{k=1..n} Sum_{i=1..k} C(n,floor((n-k)/2)+i). - Stefano Spezia, Aug 20 2019

EXAMPLE

For n = 4 consider the triangle:

45

21 24

8 13 11

2 6 7 4

0 2 4 3 1

This triangle has 45 at its apex and no other such triangle with the numbers 0 through 4 on its base has a larger apex value, so a(4) = 45.

MAPLE

a:= proc(n) return add((4*k+1)*binomial(n, k), k=0..floor((n-1)/2)) + `if`(n mod 2=0, n*binomial(n, n/2), 0):end:

seq(a(n), n=0..50);

MATHEMATICA

a[n_] := Sum[(4k+1)*Binomial[n, k], {k, 0, Floor[(n-1)/2]}] + If[EvenQ[n], n*Binomial[n, n/2], 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

PROG

(PARI) A189390(n)=sum(i=0, (n-1)\2, (4*i+1)*binomial(n, i), if(!bittest(n, 0), n*binomial(n, n\2))) \\ - M. F. Hasler, Jan 24 2012

CROSSREFS

Cf. A066411, A099325, A189162, A189391.

Sequence in context: A269754 A282425 A185003 * A099327 A004146 A275126

Adjacent sequences: A189387 A189388 A189389 * A189391 A189392 A189393

KEYWORD

easy,nonn

AUTHOR

Nathaniel Johnston, Apr 20 2011

STATUS

approved

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Last modified February 5 01:54 EST 2023. Contains 360082 sequences. (Running on oeis4.)