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 A380191 #6 Jan 25 2025 12:58:16
%S A380191 1,-1,1,-3,0,1,-12,-1,1,1,-55,-6,2,2,1,-273,-33,5,6,3,1,-1428,-182,13,
%T A380191 22,11,4,1,-7752,-1020,28,91,46,17,5,1,-43263,-5814,0,408,210,78,24,6,
%U A380191 1,-246675,-33649,-627,1938,1020,380,119,32,7,1,-1430715,-197340,-6325,9614,5187,1938,612,170,41,8,1
%N A380191 Triangle read by rows: Riordan array (2 - D(x), x * D(x)) where D(x) is g.f. of A001764.
%F A380191 T(n, k) = binomial(3*n - 2*k, n - k) * (n*k + 4*k - 3*n) / ((3*n - 2*k) * (2*n - k + 1)) if 0 <= k < n, and T(n, n) = 1 for n >= 0.
%F A380191 G.f.: (2 - D(t)) / (1 - x * t * D(t)) where D(t) is g.f. of A001764.
%F A380191 Conjecture: Sum_{i=0..n-k} binomial(2*i, i) * T(n, i+k) = A110616(n, k).
%e A380191 Triangle T(n, k) for 0 <= k <= n starts:
%e A380191 n \k : 0 1 2 3 4 5 6 7 8 9 10
%e A380191 =======================================================================
%e A380191 0 : 1
%e A380191 1 : -1 1
%e A380191 2 : -3 0 1
%e A380191 3 : -12 -1 1 1
%e A380191 4 : -55 -6 2 2 1
%e A380191 5 : -273 -33 5 6 3 1
%e A380191 6 : -1428 -182 13 22 11 4 1
%e A380191 7 : -7752 -1020 28 91 46 17 5 1
%e A380191 8 : -43263 -5814 0 408 210 78 24 6 1
%e A380191 9 : -246675 -33649 -627 1938 1020 380 119 32 7 1
%e A380191 10 : -1430715 -197340 -6325 9614 5187 1938 612 170 41 8 1
%e A380191 etc.
%o A380191 (PARI) T(n, k) = if(k==n, 1, binomial(3*n-2*k, n-k) * (n*k+4*k-3*n) / ((3*n-2*k) * (2*n-k+1)))
%Y A380191 Cf. A001764, A110616.
%K A380191 sign,easy,tabl
%O A380191 0,4
%A A380191 _Werner Schulte_, Jan 15 2025