A156644 - OEIS

%I #22 Mar 01 2021 02:02:19

%S 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,

%T -14,-14,0,14,14,6,1,-7,-20,-28,-14,14,28,20,7,1,-8,-27,-48,-42,0,42,

%U 48,27,8,1,-9,-35,-75,-90,-42,42,90,75,35,9,1

%N Mirror image of triangle A080233.

%C Inverse of A239473. Equals A007318*A167374. - _Tom Copeland_, Nov 14 2016

%H G. C. Greubel, <a href="/A156644/b156644.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

%F 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.

%F 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

%F Sum_{k=0..n} T(n, k) = A000012(n) = 1^n. - _G. C. Greubel_, Feb 28 2021

%e Triangle begins as:

%e    1;

%e    0,  1;

%e   -1,  1, 1;

%e   -2,  0, 2, 1;

%e   -3, -2, 2, 3, 1;

%e   -4, -5, 0, 5, 4, 1; ...

%t Table[Binomial[n, k] -Binomial[n, k+1], {n,0,10}, {k,0,n}]//Flatten (* _Michael De Vlieger_, Nov 24 2016 *)

%o (Sage)

%o def A156644(n,k): return ((2*k-n+1)/(k+1))*binomial(n,k)

%o flatten([[A156644(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 28 2021

%o (Magma)

%o A156644:= func< n,k | ((2*k-n+1)/(k+1))*Binomial(n,k) >;

%o [A156644(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 28 2021

%Y Cf. A024000, A080956, A097808, A120730.

%Y Cf. A007318, A167374, A239473.

%K sign,tabl

%O 0,7

%A _Philippe Deléham_, Feb 12 2009