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A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.
(Formerly M0687 N0253)
13
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

The values of a(19) and a(20) in Auluck's table on page 686 are wrong (they have been corrected here). - David W. Wilson, Mar 07 2015

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)

J. S. Birman, Letter to N. J. A. Sloane, Apr 09 1994

Erich Friedman, Illustration of initial terms

H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.

D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.

R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.

E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

FORMULA

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n-1, 1-q^k)^2 / (1-q^n) ). [Joerg Arndt, Jun 28 2013]

a(n) = sum_{m>0,k>0,2*k^2+k+2*m<=n-1} A008289(m,k)*A000041(n-k*(1+2k)-2*m-1). - [Auluck eq 29]

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-Fran├žois 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, A007293.

Row sums of triangle A259095.

Sequence in context: A098693 A122928 A200310 * A280278 A136275 A078408

Adjacent sequences:  A001521 A001522 A001523 * A001525 A001526 A001527

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected by R. K. Guy, Apr 08 1988

More terms from James A. Sellers, Feb 27 2001

STATUS

approved

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Last modified November 18 09:03 EST 2017. Contains 294861 sequences.