A259192 Triangle, such that the g.f. satisfies: A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).

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, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64, 2, 24, 100, 216, 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

Offset

0,2

Comments

Row sums = A006318, the large Schröder numbers.

Antidiagonal sums = A165941; g.f.: exp( Sum_{n>=1} 2^n*x^n/(n*(1+x^n)) ).

G.f. evaluated at y=1/2: A(x,1/2) = 1/(1-2*x).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2324, for flattened rows 0..24.

Formula

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.

Example

G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k) * x^n*y^k.
G.f.: A(x,y) = 1 + x*(2) + x^2*(2 + 4*y) +
x^3*(2 + 8*y + 4*y^2 + 8*y^3) +
x^4*(2 + 12*y + 16*y^2 + 20*y^3 + 16*y^4 + 8*y^5 + 16*y^6) +
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) +
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) +...
such that
A(x,y) = (1 + x*A(x*y,y)) / (1 - x*A(x*y,y)).
This triangle of coefficients begins:
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, 20, 64, 108, 176, 172, 208, 216, 160, 168, 144, 128, 80, 64, 32, 64;
2, 24, 100, 216, 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;
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; ...

Prog

(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) }
for(n=0, 10, for(k=0, n*(n-1)/2, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A006318, A165941.

Sequence in context: A084540 A364557 A297112 * A307313 A338042 A131999

Adjacent sequences: A259189 A259190 A259191 * A259193 A259194 A259195

Keyword

nonn,tabf,look

Author

Paul D. Hanna, Jun 21 2015

Status

approved