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

%I A108998 #25 Jan 20 2025 08:40:13
%S A108998 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 A108998 72,530,768,306,40,2,0,1,98,1072,2562,1856,482,48,2,0,1,128,1946,6968,
%U A108998 8130,3680,698,56,2,0,1,162,3264,16394,28320,20082,6432,954,64,2,0
%N A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.
%C A108998 Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.
%H A108998 Muniru A Asiru, <a href="/A108998/b108998.txt">Rows n=0..110 of antidiagonals, flattened </a>
%H A108998 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 A108998 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.
%F A108998 G.f. for coordination sequence of B_n lattice: ((Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]
%e A108998 Square array begins:
%e A108998   1,  0,    0,     0,     0,      0,      0,      0, ...
%e A108998   1,  2,    2,     2,     2,      2,      2,      2, ...
%e A108998   1,  8,   16,    24,    32,     40,     48,     56, ...
%e A108998   1, 18,   74,   170,   306,    482,    698,    954, ...
%e A108998   1, 32,  224,   768,  1856,   3680,   6432,  10304, ...
%e A108998   1, 50,  530,  2562,  8130,  20082,  42130,  78850, ...
%e A108998   1, 72, 1072,  6968, 28320,  85992, 214864, 467544, ...
%e A108998   1, 98, 1946, 16394, 83442, 307314, 907018, ...
%e A108998 Product of the g.f. of row n and (1-x)^n generates the rows of triangle A109001:
%e A108998   1;
%e A108998   1,  1;
%e A108998   1,  6,   1;
%e A108998   1, 15,  23,    1;
%e A108998   1, 28, 102,   60,    1;
%e A108998   1, 45, 290,  402,  125,   1;
%e A108998   1, 66, 655, 1596, 1167, 226, 1; ...
%o A108998 (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 A108998 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 A108998 nonn,tabl
%O A108998 0,5
%A A108998 _Paul D. Hanna_, Jun 17 2005