A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 1, 4, 10, 6, 1, 0, 0, 0, 0, 0, 3, 10, 15, 7, 1, 0, 0, 0, 0, 0, 2, 8, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 7, 19, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 5, 18, 40, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 3, 16, 41, 76, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 1, 12, 41, 86, 133, 120, 55, 12, 1

Offset

0,14

Comments

Compositions of n into k nonzero parts such that the first and last parts are 1 and the absolute difference between consecutive parts is <=1.

Row sums are A186085.

Column sums are the Motzkin numbers (A001006).

First nonzero entry in row n appears in column A055086(n).

From Joerg Arndt, Nov 06 2012: (Start)

The transposed triangle (with zeros omitted) is A129181.

For large k, the columns end in reverse([1, 1, 3, 5, 9, 14, 24, 35, ...]) for k even (cf. A053993) and reverse([1, 2, 3, 6, 10, 16, 26, 40, 60, 90, ...]) for k odd (cf. A201077).

The diagonals below the main diagonal are (apart from leading zeros), n, n*(n+1)/2, n*(n+1)*(n+2)/6, and the e-th diagonal appears to have a g.f. of the form f(x)/(1-x)^e.

(End)

Links

Alois P. Heinz, Rows n = 0..140, flattened

Joerg Arndt, the first 36 rows as Pari script.

Formula

G.f. A(x,y) satisfies: A(x,y) = 1/(1 - x*y - x^3*y^2*A(x, x*y) ). [Paul D. Hanna, Feb 22 2011]

G.f.: (formatting to make the structure apparent)

A(x,y) = 1 /

(1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 /

(1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 /

(1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 /

(1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...)))))))))))))

(continued fraction). [Paul D. Hanna]

G.f.: A(x,y) = 1/(1-x*y - x^3*y^2/(1-x^2*y - x^5*y^2/(1-x^3*y - x^7*y^2/(1 -...)))) (continued fraction). [Paul D. Hanna]

Example

Triangle begins:
1;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,1,4,10,6,1;
0,0,0,0,0,3,10,15,7,1;
0,0,0,0,0,2,8,20,21,8,1;
0,0,0,0,0,1,7,19,35,28,9,1;

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:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 24 2013

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Expand[ x*Sum[b[n-i, i+j], {j, -1, 1}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

Prog

(PARI) {T(n, k)=local(A=1+x*y); for(i=1, n, A=1/(1-x*y-x^3*y^2*subst(A, y, x*y+x*O(x^n)))); polcoeff(polcoeff(A, n, x), k, y)}
/* Paul D. Hanna */

Crossrefs

Cf. A186085 (sandpiles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086.

Cf. A186505 (antidiagonal sums).

Sequence in context: A032239 A057094 A284938 * A301345 A047998 A127841

Adjacent sequences: A186081 A186082 A186083 * A186085 A186086 A186087

Keyword

nonn,tabl

Author

Joerg Arndt, Feb 13 2011

Status

approved