A340659 -id:A340659 - cp's OEIS frontend

A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

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

Offset: 0

N. J. A. Sloane

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

			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)

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).

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.

Cf. A001522, A001523, A171604, A007293, A015128.

Row sums of triangle A259095.

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 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

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. *)

{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))^((iMichael Somos, Apr 27 2003 */

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

Corrected by R. K. Guy, Apr 08 1988

More terms from James Sellers, Feb 27 2001

A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 25, 41, 67, 105, 163, 246, 368, 540, 784, 1124, 1596, 2242, 3124, 4316, 5918, 8058, 10899, 14651, 19581, 26028, 34417, 45293, 59327, 77372, 100483, 129984, 167502, 215077, 275199, 350966, 446162, 565451, 714515, 900334, 1131370, 1417963

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

			a(3) = 4 counts the overpartitions [3], [2,1], [1,1,1], and [1',1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340659.

b:= proc(n, i, c) option remember; `if`(n=0,
     `if`(c>0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
      add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c>0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/Product_{k=1..n} (1-q^k)^2.

A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 17, 29, 49, 79, 125, 193, 293, 437, 642, 932, 1336, 1896, 2663, 3709, 5121, 7020, 9551, 12913, 17347, 23172, 30779, 40679, 53495, 70030, 91269, 118459, 153133, 197214, 253057, 323595, 412418, 523953, 663612, 838035, 1055304, 1325287, 1659969

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

Also equal to A340658(n) - A001524(n).

			a(4) = 5 counts the overpartitions [3,1], [2,2], [2,1,1], [1,1,1,1], and [1',1,1,1].

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340658, A340659.

b:= proc(n, i, c) option remember; `if`(n=0,
     `if`(c>1, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
      add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 1, 1, 0], If[i < 1, 0, b[n, i-1, c] + Sum[
     Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/ ((Product_{k=1..n} (1-q^k)^2) * (1-q^(n+1))).

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A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

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A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

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Author

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Maple

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A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula