This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A109001 #23 Jan 20 2025 08:38:50
%S A109001 1,1,1,1,6,1,1,15,23,1,1,28,102,60,1,1,45,290,402,125,1,1,66,655,1596,
%T A109001 1167,226,1,1,91,1281,4795,6155,2793,371,1,1,120,2268,12040,23750,
%U A109001 18888,5852,568,1,1,153,3732,26628,74574,91118,49380,11124,825,1,1,190,5805,53544,201810,350196,291410,114600,19629,1150,1
%N A109001 Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.
%C A109001 Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.
%H A109001 Muniru A Asiru, <a href="/A109001/b109001.txt">Rows n=0..100 of triangle, flattened</a>
%H A109001 R. Bacher, P. de la Harpe and B. Venkov, <a href="http://archive.numdam.org/item/AIF_1999__49_3_727_0">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 A109001 T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1).
%F A109001 Row sums are 2^n*(2^n - n) for n >= 0.
%F A109001 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 A109001 G.f.s of initial rows of square array A108998 are:
%e A109001 (1),
%e A109001 (1 + x)/(1-x),
%e A109001 (1 + 6*x + x^2)/(1-x)^2;
%e A109001 (1 + 15*x + 23*x^2 + x^3)/(1-x)^3;
%e A109001 (1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.
%e A109001 Triangle begins:
%e A109001 1;
%e A109001 1, 1;
%e A109001 1, 6, 1;
%e A109001 1, 15, 23, 1;
%e A109001 1, 28, 102, 60, 1;
%e A109001 1, 45, 290, 402, 125, 1;
%e A109001 1, 66, 655, 1596, 1167, 226, 1;
%e A109001 1, 91, 1281, 4795, 6155, 2793, 371, 1;
%e A109001 1, 120, 2268, 12040, 23750, 18888, 5852, 568, 1;
%e A109001 1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1;
%t A109001 T[n_, k_] := Binomial[2n+1, 2k] - 2n * Binomial[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Dec 14 2018 *)
%o A109001 (PARI) T(n,k)=binomial(2*n+1,2*k)-2*n*binomial(n-1,k-1)
%o A109001 (GAP) Flat(List([0..10],n->List([0..n],k->Binomial(2*n+1,2*k)-2*n*Binomial(n-1,k-1)))); # _Muniru A Asiru_, Dec 14 2018
%Y A109001 Cf. A108998, A108999, A109000, 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 A109001 nonn,tabl
%O A109001 0,5
%A A109001 _Paul D. Hanna_, Jun 17 2005