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A186084 Triangle T(n,k) read by rows: number of 1-dimensional sand piles (see A186085) with n grains and base length k. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 (sand piles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086.

Cf. A186505 (antidiagonal sums).

Sequence in context: A032239 A057094 A284938 * A047998 A017847 A127841

Adjacent sequences:  A186081 A186082 A186083 * A186085 A186086 A186087

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Feb 13 2011

STATUS

approved

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Last modified November 18 17:56 EST 2017. Contains 294894 sequences.