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 A385910 #16 Jul 20 2025 15:41:27
%S A385910 1,0,1,0,-1,3,0,1,-8,12,0,1,7,-49,55,0,0,9,56,-296,273,0,-2,9,14,498,
%T A385910 -1815,1428,0,0,-23,91,-288,4181,-11284,7752,0,-1,-3,-108,522,-4487,
%U A385910 33168,-70924,43263,0,1,-23,82,-579,3971,-49239,253590,-449616,246675,0,0,5,-373,2419,-6510,46017,-478291,1892593,-2869779,1430715
%N A385910 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y), read by rows.
%C A385910 Row sums form the Catalan numbers (A000108), with g.f. C(x) = 1 + x*C(x)^2.
%C A385910 Main diagonal equals A001764, with g.f. D(x) = 1 + x*D(x)^3.
%C A385910 Column 1 equals A385911.
%H A385910 Paul D. Hanna, <a href="/A385910/b385910.txt">Table of n, a(n) for n = 1..1275 (first 50 rows).</a>
%F A385910 G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies the following formulas.
%F A385910 (1) A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
%F A385910 (2) A(x,y=1) = C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F A385910 (3) T(n+1,n) = A001764(n) for n >= 0, with g.f. D(x) = 1 + x*D(x)^3.
%e A385910 G.f. A(x,y) = x + y*x^2 + (3*y^2 - y)*x^3 + (12*y^3 - 8*y^2 + y)*x^4 + (55*y^4 - 49*y^3 + 7*y^2 + y)*x^5 + (273*y^5 - 296*y^4 + 56*y^3 + 9*y^2)*x^6 + (1428*y^6 - 1815*y^5 + 498*y^4 + 14*y^3 + 9*y^2 - 2*y)*x^7 + (7752*y^7 - 11284*y^6 + 4181*y^5 - 288*y^4 + 91*y^3 - 23*y^2)*x^8 + (43263*y^8 - 70924*y^7 + 33168*y^6 - 4487*y^5 + 522*y^4 - 108*y^3 - 3*y^2 - y)*x^9 + (246675*y^9 - 449616*y^8 + 253590*y^7 - 49239*y^6 + 3971*y^5 - 579*y^4 + 82*y^3 - 23*y^2 + y)*x^10 + ...
%e A385910 where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
%e A385910 TRIANGLE.
%e A385910 Triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
%e A385910 1;
%e A385910 0, 1;
%e A385910 0, -1, 3;
%e A385910 0, 1, -8, 12;
%e A385910 0, 1, 7, -49, 55;
%e A385910 0, 0, 9, 56, -296, 273;
%e A385910 0, -2, 9, 14, 498, -1815, 1428;
%e A385910 0, 0, -23, 91, -288, 4181, -11284, 7752;
%e A385910 0, -1, -3, -108, 522, -4487, 33168, -70924, 43263;
%e A385910 0, 1, -23, 82, -579, 3971, -49239, 253590, -449616, 246675;
%e A385910 0, 0, 5, -373, 2419, -6510, 46017, -478291, 1892593, -2869779, 1430715;
%e A385910 0, 0, -2, -65, -3746, 28523, -74367, 554792, -4334344, 13891755, -18418400, 8414640; ...
%o A385910 (PARI) {T(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
%o A385910 A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
%o A385910 \\ Print the rows of the triangle
%o A385910 my(Rown); for(n=1, 12, Rown = T(n); for(k=0,n-1, print1(polcoef(Rown,k),", "));print(""))
%Y A385910 Cf. A384830 (y=-1), A376226 (y=2), A385911 (column 1), A385912 (a diagonal).
%Y A385910 Cf. A000108 (row sums), A001764 (main diagonal).
%K A385910 sign,tabl
%O A385910 1,6
%A A385910 _Paul D. Hanna_, Jul 14 2025