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

0,8

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).

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A186084, A186085.

Sequence in context: A240011 A211228 A338463 * A228693 A116676 A240575

Adjacent sequences: A186502 A186503 A186504 * A186506 A186507 A186508

Keyword

nonn

Author

Paul D. Hanna, Feb 23 2011

Status

approved