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A189162 The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n, and each number in a higher row is the sum of the two numbers directly below it.

%I A189162 #25 May 09 2023 07:14:00
%S A189162 1,3,9,24,61,148,350,808,1837,4116,9130,20056,43746,94760,204188,
%T A189162 437712,934525,1987252,4212338,8900344,18756886,39426168,82693924,
%U A189162 173071024,361567186,753984648,1569877860,3263572848,6775522852,14047800016,29091783096,60175932320
%N A189162 The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n, and each number in a higher row is the sum of the two numbers directly below it.
%C A189162 The maximum is attained by the triangle with base 1, 3, 5, ..., 2*ceiling(n/2)-1, 2*floor(n/2), ..., 6, 4, 2 (i.e., odd numbers increasing, followed by even numbers decreasing).
%H A189162 Nathaniel Johnston, <a href="/A189162/b189162.txt">Table of n, a(n) for n = 1..1000</a>
%F A189162 a(n) = 2^(n-1) + A189390(n-1).
%F A189162 D-finite with recurrence (-n+1)*a(n) +4*(n-1)*a(n-1) -12*a(n-2) +16*(-n+4)*a(n-3) +16*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 17 2021
%e A189162 For n = 5 consider the triangle:
%e A189162          61
%e A189162        29  32
%e A189162      12  17  15
%e A189162     4   8   9   6
%e A189162   1   3   5   4   2
%e A189162 This triangle has 61 at its apex and no other such triangle with the numbers 1 - 5 on its base has a larger apex value, so a(5) = 61.
%p A189162 a:=proc(n)return 2^(n-1) + add((4*k+1)*binomial(n-1,k),k=0..floor(n/2)-1) + `if`(n mod 2=1,(n-1)*binomial(n-1,(n-1)/2),0):end:
%p A189162 seq(a(n),n=1..50);
%t A189162 a[n_] := a[n] = Switch[n, 1, 1, 2, 3, 3, 9, 4, 24, _, (1/(n-1))*(4((4n-16)a[n-4] - (4n-16)a[n-3] - 3a[n-2] + (n-1)a[n-1]))];
%t A189162 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, May 09 2023, after _R. J. Mathar_ *)
%Y A189162 Cf. A066411, A099325, A189390, A189391.
%K A189162 easy,nonn
%O A189162 1,2
%A A189162 _Nathaniel Johnston_, Apr 20 2011