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 A099602 #11 Mar 29 2024 08:47:05
%S A099602 1,1,1,1,2,1,2,5,4,1,1,5,8,5,1,3,13,22,18,7,1,1,9,26,35,24,8,1,4,26,
%T A099602 70,101,84,40,10,1,1,14,61,131,160,116,49,11,1,5,45,171,363,476,400,
%U A099602 215,71,13,1,1,20,120,363,654,752,565,275,83,14,1,6,71,356,1017,1856,2282,1932
%N A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.
%C A099602 Row sums form A099603, where A099603(n) = Fibonacci(n+1)*2^floor((n+1)/2). Central coefficients of even-indexed rows form A082759, where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). Antidiagonal sums form A099604.
%C A099602 Matrix inverse equals triangle A104495, which is generated from self-convolutions of the Catalan sequence (A000108).
%F A099602 G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4).
%e A099602 Rows begin:
%e A099602 1;
%e A099602 1, 1;
%e A099602 1, 2, 1;
%e A099602 2, 5, 4, 1;
%e A099602 1, 5, 8, 5, 1;
%e A099602 3, 13, 22, 18, 7, 1;
%e A099602 1, 9, 26, 35, 24, 8, 1;
%e A099602 4, 26, 70, 101, 84, 40, 10, 1;
%e A099602 1, 14, 61, 131, 160, 116, 49, 11, 1;
%e A099602 5, 45, 171, 363, 476, 400, 215, 71, 13, 1;
%e A099602 1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1;
%e A099602 ...
%e A099602 The binomial transform of row 2 = column 2 of A027907: BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...].
%e A099602 The binomial transform of row 3 = column 3 of A027907: BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...].
%e A099602 The binomial transform of row 4 = column 4 of A027907: BINOMIAL[1,5,8,5,1] = [1,6,19,45,90,161,266,414,615,...].
%e A099602 The binomial transform of row 5 = column 5 of A027907: BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...].
%o A099602 (PARI) {T(n,k)=polcoeff(polcoeff((1+(y+1)*x-(y+1)*x^2)/(1-(y+1)*(y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}
%o A099602 (PARI) {T(n,k)=(matrix(n+1,n+1,i,j,if(i>=j,polcoeff(polcoeff( (1+x*y/(1+x))/(1+x-y^2*(1-(1+4*x+O(x^i))^(1/2))^2/4+O(y^j)),i-1,x),j-1,y)))^-1)[n+1,k+1]}
%Y A099602 Cf. A027907, A082759, A099603, A099604.
%Y A099602 Cf. A104495.
%K A099602 nonn,tabl
%O A099602 0,5
%A A099602 _Paul D. Hanna_, Oct 25 2004