The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 . 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 . Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009 LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened 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..n} T(n,k)*A000045(k) = A100334(n). 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 EXAMPLE 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; 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 *) PROG (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 CROSSREFS Cf. A063967, A167925, A199324, A202503, A202551. Sequence in context: A096874 A090046 A202551 * A199324 A287576 A035103 Adjacent sequences:  A129264 A129265 A129266 * A129268 A129269 A129270 KEYWORD sign,tabl AUTHOR Philippe Deléham, Jun 08 2007 EXTENSIONS Riordan array definition corrected by Ralf Stephan, Jan 02 2014 STATUS approved

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

Last modified April 19 11:58 EDT 2021. Contains 343114 sequences. (Running on oeis4.)