|
|
A189391
|
|
The minimum 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.
|
|
14
|
|
|
0, 1, 3, 8, 19, 44, 98, 216, 467, 1004, 2134, 4520, 9502, 19928, 41572, 86576, 179587, 372044, 768398, 1585416, 3263210, 6711176, 13775068, 28255568, 57863214, 118430584, 242061468, 494523536, 1009105372, 2058327344, 4194213448
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
This is the Riordan transform of A000217 (triangular numbers) with the Riordan matrix (of the Bell type) A053121 (inverse of the Chebyshev S Bell matrix). See the resulting formulae below. - Wolfdieter Lang, Feb 18 2017.
Conjecture: a(n) is also half the sum of the "cuts-resistance" (see A319416, A319420, A319421) of all binary vectors of length n (see Lenormand, page 4). - N. J. A. Sloane, Sep 20 2018
|
|
LINKS
|
|
|
FORMULA
|
If n even, a(n) = (n+1/2)*binomial(n,n/2) - 2^(n-1); if n odd, a(n) = ((n+1)/2)*binomial(n+1,(n+1)/2) - 2^(n-1). - N. J. A. Sloane, Nov 01 2018
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n,k).
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) + (n-5)*a(n-1) + 2*(-5*n+6)*a(n-2) + 4*(-n+8)*a(n-3) + 24*(n-3)*a(n-4) = 0. - R. J. Mathar, Jan 04 2017
G.f.: c(x^2)*Tri(x*c(x^2)), with c and Tri the g.f. of A000108 and A000217, respectively. See the explicit form of the g.f. given above by Alois P. Heinz.
(End)
|
|
EXAMPLE
|
For n = 4 consider the triangle:
....19
...8 11
..5 3 8
.4 1 2 6
3 1 0 2 4
This triangle has 19 at its apex and no other such triangle with the numbers 0 - 4 on its base has a smaller apex value, so a(4) = 19.
|
|
MAPLE
|
a:=proc(n)return add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end:
seq(a(n), n=0..50);
|
|
MATHEMATICA
|
CoefficientList[Series[(2*x+Sqrt[1-4*x^2]-1) / (2*(2*x-1)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 16 2014 *)
|
|
PROG
|
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!((2*x+Sqrt(1-4*x^2)-1)/(2*(2*x-1)^2))); // G. C. Greubel, Aug 24 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|