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A100218

Riordan array ((1-2*x)/(1-x), (1-x)).

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, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1

(list; table; graph; refs; listen; history; text; internal format)

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(2*n+1, n) = -A000007(n) = A154955(n+2).

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).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)

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, -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

Cf. A000007, A000027, A077961, A080956, A098601.

Cf. A154955, A157142, A280560, A355021.

Row sums are A100219.

Matrix inverse of A100100.