The On-Line Encyclopedia of Integer Sequences (OEIS)

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A186084

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

#46 by Joerg Arndt at Fri Mar 11 07:45:09 EST 2022

STATUS

editing

approved

#45 by Joerg Arndt at Fri Mar 11 07:45:04 EST 2022

NAME

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

CROSSREFS

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

STATUS

approved

editing

#44 by Michel Marcus at Wed Feb 18 03:43:16 EST 2015

STATUS

reviewed

approved

#43 by Joerg Arndt at Wed Feb 18 02:46:15 EST 2015

STATUS

proposed

reviewed

#42 by Jean-François Alcover at Wed Feb 18 02:35:06 EST 2015

STATUS

editing

proposed

Discussion

Wed Feb 18

02:46

Joerg Arndt: One of my favorite seqs 8-))

#41 by Jean-François Alcover at Wed Feb 18 02:35:00 EST 2015

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 *)

STATUS

approved

editing

#40 by Alois P. Heinz at Wed Jul 24 10:46:16 EDT 2013

STATUS

editing

approved

#39 by Alois P. Heinz at Wed Jul 24 10:46:04 EDT 2013

LINKS

Joerg Arndt, <a href="/A186084/a186084.gp.txt">the first 36 rows</a> as Pari script.

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

#38 by Alois P. Heinz at Wed Jul 24 10:45:27 EDT 2013

LINKS

Alois P. Heinz, <a href="/A186084/b186084.txt">Rows n = 0..140, flattened</a>

FORMULA

(continued fraction). [Paul D. Hanna]

STATUS

approved

editing

#37 by Joerg Arndt at Tue Nov 06 10:14:59 EST 2012

STATUS

editing

approved