A123148 - OEIS

A123148

Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x) for n >= 3 and 0 <= k <= n.

-1, -2, 1, -2, 2, -1, -4, 4, -2, 1, -4, 8, -6, 2, -1, -8, 12, -12, 8, -2, 1, -8, 24, -24, 16, -10, 2, -1, -16, 32, -48, 40, -20, 12, -2, 1, -16, 64, -80, 80, -60, 24, -14, 2, -1, -32, 80, -160, 160, -120, 84, -28, 16, -2, 1, -32, 160, -240, 320, -280, 168, -112, 32, -18, 2, -1

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

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = [x^k]( p(n,x) ), where p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x).

Sum_{k=0..n} T(n, k) = -1.

Sum_{k=0..n} (-1)^k * T(n,k) = -A001045(n+2).

From G. C. Greubel, Jul 17 2023: (Start)

T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k).

T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*A046854(n,k).

T(n,0) = -A016116(n+1).

T(n,1) = A171647(n).

Sum_{k=0..n} (-1)^k * abs(T(n,k)) = 1.

Sum_{k=0..floor(n/2)} T(n-k,k) = - A000034(n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = -A038754(n). (End)

EXAMPLE

The first few polynomials, p(n,x), are:

p(0,x) = -1;

p(1,x) = -2 + x;

p(2,x) = -2 + 2*x - x^2;

p(3,x) = -4 + 4*x - 2*x^2 + x^3;

p(4,x) = -4 + 8*x - 6*x^2 + 2*x^3 - x^4;

The triangle, T(n, k) = [x^k] p(n, x), begins as:

-1;

-2, 1;

-2, 2, -1;

-4, 4, -2, 1;

-4, 8, -6, 2, -1;

-8, 12, -12, 8, -2, 1;

-8, 24, -24, 16, -10, 2, -1;

MAPLE

p[0]:=-1: p[1]:=x-2: for n from 2 to 10 do p[n]:=sort(expand(-x*p[n-1]+2*p[n-2])) od: for n from 0 to 10 do seq(coeff(p[n], x, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

p[0, x]= -1; p[1, x]= x-2; p[k_, x_]:= p[k, x]= -x*p[k-1, x] + 2*p[k-2, x];

T[n_, k_]:= Coefficient[p[n, x], x, k];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Magma)

A123148:=func< n, k | (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k) >;

[A123148(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 17 2023

(SageMath)

def A123148(n, k): return (-1)^(k+1)*2^((n-k+1)//2)*binomial((n+k)//2, k)

flatten([[A123148(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 17 2023

CROSSREFS