A153764 - OEIS

%I #22 Jan 30 2022 17:34:50

%S 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,

%T 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,

%U 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

%N 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.

%C A130595*A153342 as infinite lower triangular matrices. Reflected version of A103631. Another version of A046854. Row sums are Fibonacci numbers (A000045).

%C A055830*A130595 as infinite lower triangular matrices.

%H G. C. Greubel, <a href="/A153764/b153764.txt">Table of n, a(n) for the first 45 rows</a>

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

%F 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

%F 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

%F G.f.: (1+(1-y)*x)/(1-y*x-x^2). - _Philippe Deléham_, Dec 17 2011

%F 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

%e Triangle begins:

%e   1;

%e   1, 0;

%e   1, 1, 0;

%e   1, 1, 1, 0;

%e   1, 2, 1, 1, 0;

%e   1, 2, 3, 1, 1, 0;

%e   1, 3, 3, 4, 1, 1, 0;

%e   ...

%t Table[Binomial[Floor[(n + k - 1)/2], k], {n, 0, 45}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Aug 27 2016 *)

%o (Magma) /* As triangle */ [[Binomial(Floor((n+k-1)/2),k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 28 2016

%Y Cf. A046854, A103631.

%K nonn,tabl

%O 0,12

%A _Philippe Deléham_, Jan 01 2009