A156644 Mirror image of triangle A080233.

1, 0, 1, -1, 1, 1, -2, 0, 2, 1, -3, -2, 2, 3, 1, -4, -5, 0, 5, 4, 1, -5, -9, -5, 5, 9, 5, 1, -6, -14, -14, 0, 14, 14, 6, 1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1

Offset: 0

Author

Philippe Deléham, Feb 12 2009

Keywords

• sign
• tabl

Comments

Inverse of A239473. Equals A007318*A167374. - Tom Copeland, Nov 14 2016

Examples

			Triangle begins as:
   1;
   0,  1;
  -1,  1, 1;
  -2,  0, 2, 1;
  -3, -2, 2, 3, 1;
  -4, -5, 0, 5, 4, 1; ...
		

Links

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

Crossrefs

Cf. A024000, A080956, A097808, A120730.

Cf. A007318, A167374, A239473.

Programs

Magma

A156644:= func< n,k | ((2*k-n+1)/(k+1))*Binomial(n,k) >;
[A156644(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2021

Mathematica

Table[Binomial[n, k] -Binomial[n, k+1], {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Nov 24 2016 *)

Sage

def A156644(n,k): return ((2*k-n+1)/(k+1))*binomial(n,k)
flatten([[A156644(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2021

Formula

T(n,k) = A080233(n,n-k) = (-1)^(n-k)*A097808(n,k).

T(n,k) = ((2*k-n+1)/(k+1))*binomial(n,k).

T(n,k) = T(n-1,k-1) + T(n-1,k), k>0, with T(n,0) = 1-n = A024000(n), T(n,n) = 1.

T(n,k) = binomial(n,k) - binomial(n,k+1) = Sum_{i=-k-1..k+1} (-1)^(i+1) * binomial(n,k+1+i) * binomial(n+2,k+1-i). - Mircea Merca, Apr 28 2012

Sum_{k=0..n} T(n, k) = A000012(n) = 1^n. - G. C. Greubel, Feb 28 2021