OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..49 of the triangle, flattened, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
FORMULA
Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 28 2024: (Start)
T(n, n-1) = A000027(n), n >= 1.
T(n, n-2) = -A080956(n-1), n >= 2.
T(2*n, n) = A280560(n).
T(2*n-1, n) = A157142(n-1), n >= 1.
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
From Peter Bala, Apr 28 2024: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
EXAMPLE
Triangle begins as:
1;
-1, 1;
-1, -2, 1;
-1, 0, -3, 1;
-1, 0, 2, -4, 1;
-1, 0, 0, 5, -5, 1;
-1, 0, 0, -2, 9, -6, 1;
-1, 0, 0, 0, -7, 14, -7, 1;
-1, 0, 0, 0, 2, -16, 20, -8, 1;
-1, 0, 0, 0, 0, 9, -30, 27, -9, 1;
MATHEMATICA
T[0, 0]:= 1; T[1, 1]:= 1; T[1, 0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k-1] +T[n-3, k-1]]; Table[T[n, k], {n, 0, 14}, {k, 0, n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)
PROG
(Magma)
A100218:= func< n, k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k, n-k) + Binomial(k-1, n-k-1)) >;
[A100218(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Mar 28 2024
(SageMath)
def A100218(n, k): return 1 if n==0 else (-1)^(n+k)*(binomial(k, n-k) + binomial(k-1, n-k-1))
flatten([[A100218(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Mar 28 2024
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Nov 08 2004
STATUS
approved