OFFSET
0,8
COMMENTS
Mirror image of triangle in A121314.
LINKS
Indranil Ghosh, Rows 0..125 of triangle, flattened
FORMULA
T(0,0)=1, T(n,k) = binomial(n-1+k,2k) for n >= 1.
Sum {k=0..n} T(n,k)*x^k = A000012(n), A001519(n), A001835(n), A004253(n), A001653(n), A049685(n-1), A070997(n-1), A070998(n-1), A072256(n), A078922(n), A077417(n-1), A085260(n), A001570(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Sep 26 2009
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=0. - Philippe Deléham, Feb 18 2012
G.f.: (1-x-y*x)/((1-x)^2-y*x). - Philippe Deléham, Feb 19 2012
EXAMPLE
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 3, 1, 0;
1, 6, 5, 1, 0;
1, 10, 15, 7, 1, 0;
1, 15, 35, 28, 9, 1, 0;
1, 21, 70, 84, 45, 11, 1, 0;
1, 28, 126, 210, 165, 66, 13, 1, 0;
1, 36, 210, 462, 495, 286, 91, 15, 1, 0,
1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 0;
MATHEMATICA
m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{1, 0, 1}, Table[0, {m}]], Join[{0, 1}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Sep 10 2009
STATUS
approved