A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .
1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1
Offset: 0
Examples
Triangle begins: 1; 1, 1; 0, 1, 1; -1, -1, 1, 1; -1, -3, -2, 1, 1; 0, -2, -5, -3, 1, 1; 1, 2, -2, -7, -4, 1, 1; 1, 5, 7, -1, -9, -5, 1, 1; 0, 3, 12, 15, 1, -11, -6, 1, 1; -1, -3, 3, 21, 26, 4, -13, -7, 1, 1; -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1;
Links
- G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Programs
-
Maple
T:= proc(n, k) option remember; if k<0 or k>n then 0 elif n=0 and k=0 then 1 else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020 -
Mathematica
m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *) T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *) -
Sage
@CachedFunction def T(n, k): if (k<0 or k>n): return 0 elif (n==0 and k==0): return 1 else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020
Formula
Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015
Extensions
Riordan array definition corrected by Ralf Stephan, Jan 02 2014
Comments