A153764 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 3, 4, 1, 1, 0, 1, 3, 6, 4, 5, 1, 1, 0, 1, 4, 6, 10, 5, 6, 1, 1, 0, 1, 4, 10, 10, 15, 6, 7, 1, 1, 0, 1, 5, 10, 20, 15, 21, 7, 8, 1, 1, 0, 1, 5, 15, 20, 35, 21, 28, 8, 9, 1, 1, 0, 1, 6, 15, 35, 35, 56, 28, 36, 9, 10, 1, 1, 0
Offset: 0
Examples
Triangle begins: 1; 1, 0; 1, 1, 0; 1, 1, 1, 0; 1, 2, 1, 1, 0; 1, 2, 3, 1, 1, 0; 1, 3, 3, 4, 1, 1, 0; ...
Links
- G. C. Greubel, Table of n, a(n) for the first 45 rows
Programs
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Magma
/* As triangle */ [[Binomial(Floor((n+k-1)/2),k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 28 2016
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Mathematica
Table[Binomial[Floor[(n + k - 1)/2], k], {n, 0, 45}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)
Formula
T(n,k) = binomial(floor((n+k-1)/2),k).
Sum_{k=0..n} T(n,k)*x^k = A122335(n-1), A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Dec 17 2011
Sum_{k=0..n} T(n,k)*x^(n-k) = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 17 2011
G.f.: (1+(1-y)*x)/(1-y*x-x^2). - Philippe Deléham, Dec 17 2011
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013
Comments