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A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0. 4

%I

%S 1,1,0,1,2,0,1,8,2,0,1,18,16,2,0,1,32,74,24,2,0,1,50,224,170,32,2,0,1,

%T 72,530,768,306,40,2,0,1,98,1072,2562,1856,482,48,2,0,1,128,1946,6968,

%U 8130,3680,698,56,2,0,1,162,3264,16394,28320,20082,6432,954,64,2,0

%N Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

%C Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.

%H Muniru A Asiru, <a href="/A108998/b108998.txt">Rows n=0..110 of antidiagonals, flattened </a>

%H R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

%F T(n, k) = Sum_{j=0..k} C(n+k-j-1, k-j)*(C(2*n+1, 2*j)-2*n*C(n-1, j-1)) for n >= k >= 0. G.f. for coordination sequence of B_n lattice: Sum(binomial(2*n+1, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]

%e Square array begins:

%e 1, 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 2, 2, 2, 2, 2, 2, 2, ...

%e 1, 8, 16, 24, 32, 40, 48, 56, ...

%e 1, 18, 74, 170, 306, 482, 698, 954, ...

%e 1, 32, 224, 768, 1856, 3680, 6432, 10304, ...

%e 1, 50, 530, 2562, 8130, 20082, 42130, 78850, ...

%e 1, 72, 1072, 6968, 28320, 85992, 214864, 467544, ...

%e 1, 98, 1946, 16394, 83442, 307314, 907018, ...

%e Product of the g.f. of row n and (1-x)^n generates the rows of triangle A109001:

%e 1;

%e 1, 1;

%e 1, 6, 1;

%e 1, 15, 23, 1;

%e 1, 28, 102, 60, 1;

%e 1, 45, 290, 402, 125, 1;

%e 1, 66, 655, 1596, 1167, 226, 1; ...

%o (PARI) T(n,k)=if(n<0 || k<0,0,sum(j=0,k, binomial(n+k-j-1,k-j)*(binomial(2*n+1,2*j)-2*n*binomial(n-1,j-1))))

%Y Cf. A108999 (main diagonal), A109000 (antidiagonal sums), A109001, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jun 17 2005

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Last modified June 5 12:47 EDT 2020. Contains 334840 sequences. (Running on oeis4.)