login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A100218 Riordan array ((1-2*x)/(1-x), (1-x)). 8

%I #25 Mar 28 2024 09:02:28

%S 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,

%T -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,

%U -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

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

%H G. C. Greubel, <a href="/A100218/b100218.txt">Rows n = 0..49 of the triangle, flattened, flattened</a>

%F Sum_{k=0..n} T(n, k) = A100219(n) (row sums).

%F 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

%F 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

%F From _G. C. Greubel_, Mar 28 2024: (Start)

%F T(n, n-1) = A000027(n), n >= 1.

%F T(n, n-2) = -A080956(n-1), n >= 2.

%F T(2*n, n) = A280560(n).

%F T(2*n-1, n) = A157142(n-1), n >= 1.

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

%F T(3*n, n) = T(4*n, n) = A000007(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).

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

%e Triangle begins as:

%e 1;

%e -1, 1;

%e -1, -2, 1;

%e -1, 0, -3, 1;

%e -1, 0, 2, -4, 1;

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

%e -1, 0, 0, -2, 9, -6, 1;

%e -1, 0, 0, 0, -7, 14, -7, 1;

%e -1, 0, 0, 0, 2, -16, 20, -8, 1;

%e -1, 0, 0, 0, 0, 9, -30, 27, -9, 1;

%t 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 *)

%o (Magma)

%o A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;

%o [A100218(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Mar 28 2024

%o (SageMath)

%o def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))

%o flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Mar 28 2024

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

%Y Cf. A154955, A157142, A280560, A355021.

%Y Row sums are A100219.

%Y Matrix inverse of A100100.

%Y Apart from signs, same as A098599.

%Y Very similar to triangle A111125.

%K easy,sign,tabl,changed

%O 0,5

%A _Paul Barry_, Nov 08 2004

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)