cp's OEIS Frontend

A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

%I A121314 #24 Jan 21 2020 21:19:40
%S A121314 1,0,1,0,1,1,0,1,3,1,0,1,5,6,1,0,1,7,15,10,1,0,1,9,28,35,15,1,0,1,11,
%T A121314 45,84,70,21,1,0,1,13,66,165,210,126,28,1,0,1,15,91,286,495,462,210,
%U A121314 36,1
%N A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
%C A121314 A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...
%C A121314 Mirror image of triangle in A165253.
%H A121314 Alois P. Heinz, <a href="/A121314/b121314.txt">Rows n = 0..140, flattened</a>
%H A121314 F. Yano and H. Yoshida, <a href="https://doi.org/10.1016/j.disc.2007.03.050">Some set partition statistics in non-crossing partitions and generating functions</a>, Discr. Math., 307 (2007), 3147-3160.
%F A121314 T(0,0)=1; T(n,0)=0 for n > 0; T(n+1,k+1) = binomial(2*n-k,k)for n >= 0 and k >= 0.
%F A121314 Sum_{k=0..n} T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively.
%F A121314 Sum_{k=0..n} 2^k*T(n,k) = (4^n+2)/3.
%F A121314 Sum_{k=0..n} 2^(n-k)*T(n,k) = A001835(n).
%F A121314 Sum_{k=0..n} 3^k*4^(n-k)*T(n,k) = A054879(n). - _Philippe Deléham_, Aug 26 2006
%F A121314 Sum_{k=0..n} T(n,k)*(-1)^k*2^(3n-2k) = A143126(n). - _Philippe Deléham_, Oct 31 2008
%F A121314 Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A138340(n)/4^n. - _Philippe Deléham_, Nov 01 2008
%F A121314 G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - _Philippe Deléham_, Nov 01 2011
%F A121314 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0. - _Philippe Deléham_, Feb 19 2012
%e A121314 Triangle begins
%e A121314   1;
%e A121314   0,  1;
%e A121314   0,  1,  1;
%e A121314   0,  1,  3,  1;
%e A121314   0,  1,  5,  6,  1;
%e A121314   0,  1,  7, 15, 10,  1;
%e A121314   0,  1,  9, 28, 35, 15,  1;
%e A121314   0,  1, 11, 45, 84, 70, 21,  1;
%Y A121314 Cf. A054142, A165253.
%K A121314 nonn,tabl
%O A121314 0,9
%A A121314 _Philippe Deléham_, Aug 25 2006