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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) 16
 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 Also the number of overpartitions of n having more overlined parts than non-overlined parts. For example, a(5) = 5 counts the overpartitions [5'], [4',1'], [3',2'], [3',1',1] and [2',2,1']. - Jeremy Lovejoy, Jan 15 2021 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 Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe) 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 Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020. 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. B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp. 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] From Vaclav Kotesovec, Mar 03 2020: (Start) Pi * sqrt(2/3) <= n^(-1/2)*log(a(n)) <= Pi * sqrt(5/6). [Auluck, 1951] log(a(n)) ~ 2*Pi*sqrt(n/5). [Wright, 1971] a(n) ~ exp(2*Pi*sqrt(n/5)) / (sqrt(2) * 5^(3/4) * (1 + sqrt(5)) * n). (End) a(n) = A143184(n) - A340659(n). - Vaclav Kotesovec, Jun 06 2021 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 # second Maple program: b:= proc(n, i) option remember; `if`(i>n, 0, `if`( irem(n, i)=0, 1, 0)+add(j*b(n-i*j, i+1), j=1..n/i)) end: a:= n-> `if`(n=0, 1, b(n, 1)): seq(a(n), n=0..100); # Alois P. Heinz, Oct 03 2018 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

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Last modified February 29 00:39 EST 2024. Contains 370400 sequences. (Running on oeis4.)