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A165253 Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

%I A165253 #23 Feb 19 2020 11:33:21
%S A165253 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,
%T A165253 84,45,11,1,0,1,28,126,210,165,66,13,1,0,1,36,210,462,495,286,91,15,1,
%U A165253 0,1,45,330,924,1287,1001,455,120,17,1,0,1,55,495,1716,3003,3003,1820,680
%N A165253 Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C A165253 Mirror image of triangle in A121314.
%H A165253 Indranil Ghosh, <a href="/A165253/b165253.txt">Rows 0..125 of triangle, flattened</a>
%F A165253 T(0,0)=1, T(n,k) = binomial(n-1+k,2k) for n >= 1.
%F A165253 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.
%F A165253 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
%F A165253 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
%F A165253 G.f.: (1-x-y*x)/((1-x)^2-y*x). - _Philippe Deléham_, Feb 19 2012
%e A165253 Triangle begins:
%e A165253   1;
%e A165253   1,    0;
%e A165253   1,    1,    0;
%e A165253   1,    3,    1,    0;
%e A165253   1,    6,    5,    1,    0;
%e A165253   1,   10,   15,    7,    1,    0;
%e A165253   1,   15,   35,   28,    9,    1,    0;
%e A165253   1,   21,   70,   84,   45,   11,    1,    0;
%e A165253   1,   28,  126,  210,  165,   66,   13,    1,    0;
%e A165253   1,   36,  210,  462,  495,  286,   91,   15,    1,    0,
%e A165253   1,   45,  330,  924, 1287, 1001,  455,  120,   17,    1,    0;
%t A165253 m = 13;
%t A165253 (* DELTA is defined in A084938 *)
%t A165253 DELTA[Join[{1, 0, 1}, Table[0, {m}]], Join[{0, 1}, Table[0, {m}]], m] // Flatten (* _Jean-François Alcover_, Feb 19 2020 *)
%Y A165253 Cf. A054142, A085478, A121314.
%K A165253 nonn,tabl
%O A165253 0,8
%A A165253 _Philippe Deléham_, Sep 10 2009