|
| |
|
|
A001524
|
|
Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.
(Formerly M0687 N0253)
|
|
8
|
|
|
|
1, 1, 1, 2, 3, 5, 8, 12, 18, 26, 38, 53, 75, 103, 142, 192, 260, 346, 461, 607, 797, 1038, 1348, 1738, 2234, 2856, 3638, 4614, 5832, 7342, 9214, 11525, 14369, 17863, 22142, 27371, 33744, 41498, 50903, 62299, 76066, 92676, 112666, 136696, 165507, 200018
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Also n-stacks with strictly receding left wall.
Weakly unimodal compositions such that each up-step is by at most 1 (and first part 1). By dropping the requirement for weak unimodality one obtains A005169. [Joerg Arndt, Dec 09 2012]
|
|
|
REFERENCES
|
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 0..1000
Erich Friedman, Illustration of initial terms
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice.
|
|
|
EXAMPLE
|
For a(6)=8 we have the following stacks:
..x
.xx .xx. ..xx .x... ..x.. ...x. ....x
xxx xxxx xxxx xxxxx xxxxx xxxxx xxxxx xxxxxx
From Franklin T. Adams-Watters, Jan 18 2007: (Start)
For a(7) = 12 we have the following stacks:
..x. ...x
.xx. ..xx .xxx .xx.. ..xx. ...xx
xxxx xxxx xxxx xxxxx xxxxx xxxxx
and
.x.... ..x... ...x.. ....x. .....x
xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxxx
(End)
|
|
|
MAPLE
|
s := 1+sum(z^(n*(n+1)/2)/((1-z^(n))*product((1-z^i), i=1..n-1)^2), n=1..50): s2 := series(s, z, 300): for j from 1 to 100 do printf(`%d, `, coeff(s2, z, j)) od:
# James A. Sellers, Feb 27 2001
|
|
|
MATHEMATICA
|
m = 45; CoefficientList[ Series[Sum[ z^(n*(n+1)/2)/((1-z^(n))*Product[(1-z^i), {i, 1, n-1}]^2), {n, 1, m}], {z, 0, m}], z] // Prepend[Rest[#], 1] &
(* Jean-François Alcover, May 19 2011, after Maple prog. *)
|
|
|
PROG
|
(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=0, (sqrt(8*n + 1) - 1) / 2, x^((k^2 + k) / 2) / prod( i=1, k, (1 - x^i + x * O(x^n))^((i<k) + 1))), n))} /* Michael Somos, Apr 27 2003 */
|
|
|
CROSSREFS
|
Cf. A001522, A001523, A171604
Sequence in context: A098693 A122928 A200310 * A136275 A078408 A007478
Adjacent sequences: A001521 A001522 A001523 * A001525 A001526 A001527
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
EXTENSIONS
|
More terms from James A. Sellers, Feb 27 2001
|
|
|
STATUS
|
approved
|
| |
|
|
