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 A379907 #16 Jan 06 2025 13:15:49
%S A379907 1,0,1,1,1,1,1,2,2,1,3,4,4,3,1,6,9,9,7,4,1,15,21,21,17,11,5,1,36,51,
%T A379907 51,42,29,16,6,1,91,127,127,106,76,46,22,7,1,232,323,323,272,200,128,
%U A379907 69,29,8,1,603,835,835,708,530,352,204,99,37,9,1,1585,2188,2188,1865,1415,965,587,311,137,46,10,1
%N A379907 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} (-1)^(n - k - i) * binomial(n - k, i) * binomial(k + 2*i, i) * (k + 1) / (k + 1 + i).
%C A379907 Conjecture: Let A = (g(t), f(t)) and B = (u(t), v(t)) be (triangular) Riordan arrays with A(n, k) = [t^n](g(t)*(f(t))^k) and B(n, k) = [t^n](u(t)*(v(t))^k). Then T = (g(t)*u(f(t)), v(f(t))*t/f(t)) is the Riordan array with T(n, k) = [t^n](g(t)*u(f(t))*(v(f(t))*t/f(t))^k) = Sum_{i=0..n-k} A(n-k, i) * B(k+i, k) for 0 <= k <= n.
%F A379907 Riordan array (C(t/(1+t)) / (1+t), t * C(t/(1+t))) where C(x) is g.f. of A000108.
%F A379907 Riordan array ((1 + t - sqrt(1 - 2*t - 3*t^2))/(2*t*(1 + t)), (1 + t - sqrt(1-2*t-3*t^2))/2).
%F A379907 G.f.: 2/(sqrt((1 - 3*t)*(t + 1)) - 2*(t + 1)*t*x + t + 1).
%F A379907 Conjecture: T(n, k) = T(n, k-1) + T(n-1, k-1) - T(n-1, k-2) - T(n-2, k-2) for 2 <= k <= n.
%F A379907 T(n, k) = (-1)^(k-n)*hypergeom([k-n, k/2+1, (k+1)/2], [1, k + 2], 4). - _Peter Luschny_, Jan 06 2025
%e A379907 Triangle T(n, k) for 0 <= k <= n starts:
%e A379907 n \k : 0 1 2 3 4 5 6 7 8 9 10 11
%e A379907 ====================================================================
%e A379907 0 : 1
%e A379907 1 : 0 1
%e A379907 2 : 1 1 1
%e A379907 3 : 1 2 2 1
%e A379907 4 : 3 4 4 3 1
%e A379907 5 : 6 9 9 7 4 1
%e A379907 6 : 15 21 21 17 11 5 1
%e A379907 7 : 36 51 51 42 29 16 6 1
%e A379907 8 : 91 127 127 106 76 46 22 7 1
%e A379907 9 : 232 323 323 272 200 128 69 29 8 1
%e A379907 10 : 603 835 835 708 530 352 204 99 37 9 1
%e A379907 11 : 1585 2188 2188 1865 1415 965 587 311 137 46 10 1
%e A379907 etc.
%p A379907 gf := 2/(sqrt((1-3*t)*(t+1)) - 2*(t+1)*t*x + t+1): ser := simplify(series(gf,t,12)):
%p A379907 ct := n -> coeff(ser,t,n): row := n -> local k; seq(coeff(ct(n), x, k), k = 0..n):
%p A379907 seq(row(n), n = 0..11); # _Peter Luschny_, Jan 05 2025
%o A379907 (PARI) T(n,k) = sum(i=0,n-k,(-1)^(n-k-i)*binomial(n-k,i)*binomial(k+2*i,i)*(k+1)/(k+1+i))
%o A379907 (PARI) T(n,k)=polcoef(polcoef(2/(sqrt((1-3*t)*(1+t))+(1+t)*(1-2*x*t))+x*O(x^k),k,x)+t*O(t^n),n,t);
%o A379907 m=matrix(15,15,n,k,if(k>n,0,T(n-1,k-1)))
%Y A379907 Cf. A005043 (column 0), A001006 (column 1 and 2), A102071 (column 3).
%Y A379907 Cf. A000108, A342912 (row sums), A379824 (alternating row sums), A379823 (central terms).
%K A379907 nonn,easy,tabl
%O A379907 0,8
%A A379907 _Werner Schulte_, Jan 05 2025