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A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A205813 #21 May 13 2025 22:01:39
%S A205813 1,0,1,0,2,1,0,6,4,1,0,20,16,6,1,0,70,64,30,8,1,0,252,256,140,48,10,1,
%T A205813 0,924,1024,630,256,70,12,1,0,3432,4096,2772,1280,420,96,14,1,0,12870,
%U A205813 16384,12012,6144,2310,640,126,16,1
%N A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A205813 Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).
%C A205813 T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - _Peter Luschny_, Oct 19 2022
%F A205813 T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).
%F A205813 Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.
%F A205813 G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).
%F A205813 Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.
%e A205813 Triangle begins:
%e A205813   1;
%e A205813   0,   1;
%e A205813   0,   2,   1;
%e A205813   0,   6,   4,   1;
%e A205813   0,  20,  16,   6,   1;
%e A205813   0,  70,  64,  30,   8,   1;
%e A205813   0, 252, 256, 140,  48,  10,   1;
%p A205813 # Uses function PMatrix from A357368.
%p A205813 PMatrix(10, n -> binomial(2*(n-1), n-1)); # _Peter Luschny_, Oct 19 2022
%Y A205813 Cf. A054335 and columns listed there.
%K A205813 easy,nonn,tabl
%O A205813 0,5
%A A205813 _Philippe Deléham_, Feb 01 2012