A186505 Antidiagonal sums of triangle A186084.

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 8, 9, 14, 18, 25, 34, 46, 64, 86, 119, 162, 222, 304, 416, 571, 780, 1071, 1466, 2010, 2754, 3775, 5175, 7092, 9724, 13329, 18274, 25052, 34347, 47091, 64562, 88522, 121369, 166411, 228168, 312848, 428959, 588163

Offset: 0

Author

Paul D. Hanna, Feb 23 2011

Keywords

• nonn

Comments

Triangle A186084 is the number of 1-dimensional sandpiles with n grains and base length k.

The g.f. T(x,y) of triangle A186084 satisfies: T(x,y) = 1/(1 - x*y - x^3*y^2*T(x,x*y)); therefore, the g.f. of this sequence is T(x,x).

Examples

			G.f.: 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A186084, A186085.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
    end:
a:= n-> add(coeff(b(n-k, 1), x, k), k=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Jul 24 2013

Mathematica

m = 100;
f[i_] := If[i == 0, 1, -x^(2i+3)];
g[i_] := 1 - x^(i+2);
ContinuedFractionK[f[i], g[i], {i, 0, Sqrt[m] // Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Oct 14 2019, after Sergei N. Gladkovskii *)

PARI

{a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(Txy,y,x), n, x)}

PARI

N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );
gf = 1/x^3 - (Q(0) + 1)/x^2;
Vec(gf) \\ Joerg Arndt, May 07 2013

Formula

G.f.: (1 - x/(1 - 1/B(x)))/x^3 where B(x) equals the g.f. of the row sums of triangle A186084.

G.f.: 1/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1-x^5 - x^11/(1-x^6 - x^13/(1-...)))))) (continued fraction).

G.f.: 1/(1-x^2/(1-x^3/(1-x^7/(1-x^4/(1-x^5/(1-x^11/(1-x^6/(1 -x^7/(1-x^15/(1-...)))))))))) (continued fraction).

G.f.: 1/x^3 - (Q(0) + 1)/x^2, where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013

a(n) ~ c * d^n, where d = 1.3712018040437285..., c = 0.154355235026898... . - Vaclav Kotesovec, Sep 10 2014