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 A259192 #28 Dec 30 2017 10:01:28
%S A259192 1,2,2,4,2,8,4,8,2,12,16,20,16,8,16,2,16,36,48,68,40,64,40,32,16,32,2,
%T A259192 20,64,108,176,172,208,216,160,168,144,128,80,64,32,64,2,24,100,216,
%U A259192 388,528,612,784,704,792,672,728,576,560,384,464,288,256,160,128,64,128,2,28,144,388,784,1300,1696,2316,2544,2864,2976,3000,3024,2856,2560,2400,2416,1856,1776,1408,1248,1024,928,576,512,320,256,128,256
%N A259192 Triangle, such that the g.f. satisfies: A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
%C A259192 Row sums = A006318, the large Schröder numbers.
%C A259192 Antidiagonal sums = A165941; g.f.: exp( Sum_{n>=1} 2^n*x^n/(n*(1+x^n)) ).
%C A259192 G.f. evaluated at y=1/2: A(x,1/2) = 1/(1-2*x).
%H A259192 Paul D. Hanna, <a href="/A259192/b259192.txt">Table of n, a(n) for n = 0..2324, for flattened rows 0..24.</a>
%F A259192 G.f.: A(x,y) = -1 + 2/(1+x - 2*x/(1+x*y - 2*x*y/(1+x*y^2 - 2*x*y^2/(1+x*y^3 - 2*x*y^3/(1+x*y^4 - 2*x*y^4/(1+x*y^5 - 2*x*y^5/(1+x*y^6 - 2*x*y^6/(1+x*y^7 -...)))))))), a continued fraction.
%e A259192 G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k) * x^n*y^k.
%e A259192 G.f.: A(x,y) = 1 + x*(2) + x^2*(2 + 4*y) +
%e A259192 x^3*(2 + 8*y + 4*y^2 + 8*y^3) +
%e A259192 x^4*(2 + 12*y + 16*y^2 + 20*y^3 + 16*y^4 + 8*y^5 + 16*y^6) +
%e A259192 x^5*(2 + 16*y + 36*y^2 + 48*y^3 + 68*y^4 + 40*y^5 + 64*y^6 + 40*y^7 + 32*y^8 + 16*y^9 + 32*y^10) +
%e A259192 x^6*(2 + 20*y + 64*y^2 + 108*y^3 + 176*y^4 + 172*y^5 + 208*y^6 + 216*y^7 + 160*y^8 + 168*y^9 + 144*y^10 + 128*y^11 + 80*y^12 + 64*y^13 + 32*y^14 + 64*y^15) +...
%e A259192 such that
%e A259192 A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
%e A259192 This triangle of coefficients begins:
%e A259192 1;
%e A259192 2;
%e A259192 2, 4;
%e A259192 2, 8, 4, 8;
%e A259192 2, 12, 16, 20, 16, 8, 16;
%e A259192 2, 16, 36, 48, 68, 40, 64, 40, 32, 16, 32;
%e A259192 2, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64;
%e A259192 2, 24, 100, 216, 388, 528, 612, 784, 704, 792, 672, 728, 576, 560, 384, 464, 288, 256, 160, 128, 64, 128;
%e A259192 2, 28, 144, 388, 784, 1300, 1696, 2316, 2544, 2864, 2976, 3000, 3024, 2856, 2560, 2400, 2416, 1856, 1776, 1408, 1248, 1024, 928, 576, 512, 320, 256, 128, 256;
%e A259192 2, 32, 196, 640, 1476, 2808, 4260, 6104, 7844, 9216, 10816, 11264, 12512, 12424, 12608, 11784, 12384, 10848, 10880, 9328, 8992, 7888, 7488, 5952, 5856, 4352, 4064, 3072, 3008, 2048, 1856, 1152, 1024, 640, 512, 256, 512; ...
%o A259192 (PARI) {T(n,k) = local(A=1+2*x); for(i=1,n, A = (1 + x*subst(A,x,x*y))/(1 - x*subst(A,x,x*y +x*O(x^n))) ); polcoeff( polcoeff(A,n,x) ,k,y) }
%o A259192 for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))
%Y A259192 Cf. A006318, A165941.
%K A259192 nonn,tabf,look
%O A259192 0,2
%A A259192 _Paul D. Hanna_, Jun 21 2015