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%I #28 Dec 30 2017 10:01:28
%S 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 20,64,108,176,172,208,216,160,168,144,128,80,64,32,64,2,24,100,216,
%U 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 Triangle, such that the g.f. satisfies: A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
%C Row sums = A006318, the large Schröder numbers.
%C Antidiagonal sums = A165941; g.f.: exp( Sum_{n>=1} 2^n*x^n/(n*(1+x^n)) ).
%C G.f. evaluated at y=1/2: A(x,1/2) = 1/(1-2*x).
%H Paul D. Hanna, <a href="/A259192/b259192.txt">Table of n, a(n) for n = 0..2324, for flattened rows 0..24.</a>
%F 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 G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k) * x^n*y^k.
%e G.f.: A(x,y) = 1 + x*(2) + x^2*(2 + 4*y) +
%e x^3*(2 + 8*y + 4*y^2 + 8*y^3) +
%e x^4*(2 + 12*y + 16*y^2 + 20*y^3 + 16*y^4 + 8*y^5 + 16*y^6) +
%e 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 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 such that
%e A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
%e This triangle of coefficients begins:
%e 1;
%e 2;
%e 2, 4;
%e 2, 8, 4, 8;
%e 2, 12, 16, 20, 16, 8, 16;
%e 2, 16, 36, 48, 68, 40, 64, 40, 32, 16, 32;
%e 2, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64;
%e 2, 24, 100, 216, 388, 528, 612, 784, 704, 792, 672, 728, 576, 560, 384, 464, 288, 256, 160, 128, 64, 128;
%e 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 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 (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 for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))
%Y Cf. A006318, A165941.
%K nonn,tabf,look
%O 0,2
%A _Paul D. Hanna_, Jun 21 2015
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