A247492 - OEIS

%I #15 Oct 23 2021 10:54:56

%S 1,-1,2,1,0,3,-1,0,2,4,1,0,0,5,5,-1,0,0,2,9,6,1,0,0,0,7,14,7,-1,0,0,0,

%T 2,16,20,8,1,0,0,0,0,9,30,27,9,-1,0,0,0,0,2,25,50,35,10,1,0,0,0,0,0,

%U 11,55,77,44,11,-1,0,0,0,0,0,2,36,105,112,54,12

%N Triangle read by rows: T(n, k) = binomial(k-1, n-k)*(n+1)/(n+1-k), 0 <= k <= n.

%F Sum_{k = 0..n} T(n, k) = A001350(n+1).

%F G.f.: (x^2*y + 1)/((x^4 + 2*x^3 + x^2)*y^2 + (-x^3 - 3*x^2 - 2*x)*y + x + 1). Or: (x^2*y + 1)/((x + 1)*(x*y - 1)*(x^2*y + x*y - 1)). - _Vladimir Kruchinin_, Oct 23 2021

%e [0]  1;

%e [1] -1, 2;

%e [2]  1, 0, 3;

%e [3] -1, 0, 2, 4;

%e [4]  1, 0, 0, 5, 5;

%e [5] -1, 0, 0, 2, 9, 6;

%e [6]  1, 0, 0, 0, 7, 14, 7;

%e .

%e Taylor series: 1 + x*(2*y - 1) + x^2*(3*y^2 + 1) + x^3*(4*y^3 + 2*y^2 - 1) + x^4*(5*y^4 + 5*y^3 + 1) + O(x^5).

%p T := (n, k) -> (n+1)*binomial(k-1, n-k)/(n+1-k);

%p for n from 0 to 11 do seq(T(n,k), k=0..n) od;

%Y Cf. A001350 (row sums), A098599, A100218.

%K sign,tabl

%O 0,3

%A _Peter Luschny_, Oct 01 2014