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%I M0687 N0253 #95 Jun 06 2021 05:49:39
%S 1,1,1,2,3,5,8,12,18,26,38,53,75,103,142,192,260,346,461,607,797,1038,
%T 1348,1738,2234,2856,3638,4614,5832,7342,9214,11525,14369,17863,22142,
%U 27371,33744,41498,50903,62299,76066,92676,112666,136696,165507,200018
%N Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.
%C Also n-stacks with strictly receding left wall.
%C 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
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001524/b001524.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100027134">On some new types of partitions associated with generalized Ferrers graphs</a>, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
%H F. C. Auluck, <a href="/A001524/a001524.pdf">On some new types of partitions associated with generalized Ferrers graphs</a> (annotated scanned copy)
%H J. S. Birman, <a href="/A007293/a007293.pdf">Letter to N. J. A. Sloane, Apr 09 1994</a>
%H Sergi Elizalde, <a href="https://arxiv.org/abs/2008.05669">Symmetric peaks and symmetric valleys in Dyck paths</a>, arXiv:2008.05669 [math.CO], 2020.
%H Erich Friedman, <a href="/A001524/a001524.gif">Illustration of initial terms</a>
%H H. W. Gould, R. K. Guy, and N. J. A. Sloane, <a href="/A005169/a005169_5.pdf">Correspondence</a>, 1987.
%H D. Gouyou-Beauchamps and P. Leroux, <a href="http://www.arXiv.org/abs/math.CO/0403168">Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice</a>, arXiv:math/0403168 [math.CO], 2004.
%H R. K. Guy, <a href="/A259095/a259095.pdf">Letter to N. J. A. Sloane, Apr 08 1988</a> (annotated scanned copy, included with permission)
%H R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.
%H R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1016/j.ejc.2020.103159">Parity bias in partitions</a>, European J. Combin., 89 (2020), 103159, 19 pp.
%H E. M. Wright, <a href="http://qjmath.oxfordjournals.org/content/23/2/153.extract">Stacks, III</a>, Quart. J. Math. Oxford, 23 (1972), 153-158.
%F 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]
%F 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]
%F From _Vaclav Kotesovec_, Mar 03 2020: (Start)
%F Pi * sqrt(2/3) <= n^(-1/2)*log(a(n)) <= Pi * sqrt(5/6). [Auluck, 1951]
%F log(a(n)) ~ 2*Pi*sqrt(n/5). [Wright, 1971]
%F a(n) ~ exp(2*Pi*sqrt(n/5)) / (sqrt(2) * 5^(3/4) * (1 + sqrt(5)) * n). (End)
%F a(n) = A143184(n) - A340659(n). - _Vaclav Kotesovec_, Jun 06 2021
%e For a(6)=8 we have the following stacks:
%e ..x
%e .xx .xx. ..xx .x... ..x.. ...x. ....x
%e xxx xxxx xxxx xxxxx xxxxx xxxxx xxxxx xxxxxx
%e From _Franklin T. Adams-Watters_, Jan 18 2007: (Start)
%e For a(7) = 12 we have the following stacks:
%e ..x. ...x
%e .xx. ..xx .xxx .xx.. ..xx. ...xx
%e xxxx xxxx xxxx xxxxx xxxxx xxxxx
%e and
%e .x.... ..x... ...x.. ....x. .....x
%e xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxxx
%e (End)
%p 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
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(i>n, 0, `if`(
%p irem(n, i)=0, 1, 0)+add(j*b(n-i*j, i+1), j=1..n/i))
%p end:
%p a:= n-> `if`(n=0, 1, b(n, 1)):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Oct 03 2018
%t 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] &
%t (* _Jean-François Alcover_, May 19 2011, after Maple prog. *)
%o (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 */
%Y Cf. A001522, A001523, A171604, A007293, A015128.
%Y Row sums of triangle A259095.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_
%E Corrected by _R. K. Guy_, Apr 08 1988
%E More terms from _James A. Sellers_, Feb 27 2001
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