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%I #20 Sep 07 2024 21:10:06
%S 1,2,1,0,3,1,0,2,4,1,0,0,5,5,1,0,0,2,9,6,1,0,0,0,7,14,7,1,0,0,0,2,16,
%T 20,8,1,0,0,0,0,9,30,27,9,1,0,0,0,0,2,25,50,35,10,1,0,0,0,0,0,11,55,
%U 77,44,11,1,0,0,0,0,0,2,36,105,112,54,12,1,0,0,0,0,0,0,13,91,182,156,65,13,1
%N Riordan array (1+2x,x(1+x)).
%C Row sums are Lucas numbers A000204. Diagonal sums are A007307(n+1). Inverse is (-1)^(n-k)A092392(n,k). Product with Pascal triangle (1/(1-x),x/(1-x)) is A111125.
%H R. Tauraso, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Tauraso/tauraso18.html">Edge cover time for regular graphs</a>, JIS 11 (2008) 08.4.4.
%F T(n, k) = C(k, n-k) + 2*C(k, n-k-1).
%F T(n, k) = Sum_{j = 0..n} (-1)^(n-j)*C(n, j)*C(j+k, 2*k)*(2*j+1)/(2*k+1).
%F From _Peter Bala_, Sep 10 2021: (Start)
%F T(n,k) = T(n-1,k-1) + T(n-2,k-1) with boundary conditions T(n,n) = 1, T(1,0) = 2 and T(n,k) = 0 for k < 0 or k > n.
%F The entries in row n, read in reverse order, are the coefficients in the n-th degree Taylor polynomial of (1 + x*c(-x))^(n+1) at x = 0, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%e Triangle begins
%e 1;
%e 2, 1;
%e 0, 3, 1;
%e 0, 2, 4, 1;
%e 0, 0, 5, 5, 1;
%e 0, 0, 2, 9, 6, 1;
%e 0, 0, 0, 7, 14, 7, 1;
%e 0, 0, 0, 2, 16, 20, 8, 1;
%e Row 4: (1 + x*c(-x))^5 = 1 + 5*x + 5*x^2 + O(x^5). - _Peter Bala_, Sep 10 2021
%Y Cf. A000108, A000204, A007307, A092392, A111125.
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Oct 18 2005