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.

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

Offset: 0

Author

Nathaniel Johnston, Apr 20 2011

Keywords

• easy
• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A066411, A099325, A189162, A189391.

Programs

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 *)

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

Python

from math import comb
def A189390(n): return sum(((k<<2)|1)*comb(n,k) for k in range(n+1>>1))+(0 if n&1 else n*comb(n,n>>1)) # Chai Wah Wu, Oct 28 2024

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