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A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).

%I A157458 #25 May 28 2020 03:02:15
%S A157458 0,1,1,1,2,1,1,3,3,1,1,3,4,3,1,1,3,5,5,3,1,1,3,5,6,5,3,1,1,3,5,7,7,5,
%T A157458 3,1,1,3,5,7,8,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1,3,5,7,9,10,9,7,5,3,1,1,3,
%U A157458 5,7,9,11,11,9,7,5,3,1,1,3,5,7,9,11,12,11,9,7,5,3,1,1,3,5,7,9,11,13,13,11,9,7,5,3,1
%N A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).
%C A157458 The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.
%C A157458 Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - _N. J. A. Sloane_, Aug 27 2009
%H A157458 Nathaniel Johnston, <a href="/A157458/b157458.txt">Table of n, a(n) for n = 0..10000</a>
%F A157458 T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).
%F A157458 From _Yu-Sheng Chang_, May 19 2020: (Start)
%F A157458 O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).
%F A157458 T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)
%e A157458 Triangle begins as:
%e A157458   0;
%e A157458   1, 1;
%e A157458   1, 2, 1;
%e A157458   1, 3, 3, 1;
%e A157458   1, 3, 4, 3, 1;
%e A157458   1, 3, 5, 5, 3, 1;
%e A157458   1, 3, 5, 6, 5, 3, 1;
%e A157458   1, 3, 5, 7, 7, 5, 3, 1;
%e A157458   1, 3, 5, 7, 8, 7, 5, 3, 1;
%e A157458   1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
%e A157458   1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1;
%p A157458 T := proc(m,n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m,n),m=0..n),n=0..14); # _Nathaniel Johnston_, Apr 29 2011
%t A157458 T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten
%Y A157458 Cf. A003983, A157457.
%K A157458 nonn,tabl,easy
%O A157458 0,5
%A A157458 _Roger L. Bagula_, Mar 01 2009
%E A157458 Edited by _N. J. A. Sloane_, Aug 27 2009
%E A157458 More terms from and partially edited by _G. C. Greubel_, May 21 2020