A143184 -id:A143184 - 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

A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

Original entry on oeis.org

1, 0, 1, 2, 3, 5, 7, 11, 15, 23, 31, 45, 61, 85, 114, 156, 206, 276, 363, 477, 621, 808, 1041, 1339, 1713, 2182, 2769, 3501, 4409, 5534, 6927, 8635, 10741, 13316, 16467, 20303, 24980, 30643, 37518, 45815, 55836, 67889, 82395, 99772, 120609, 145501, 175229, 210637

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340658.

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 *)
nmax = 50; CoefficientList[Series[1 + Sum[x^(j*(j+1)/2 + j) / QPochhammer[x, x, j]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2021 *)

G.f.: Sum_{n>=0} q^(n*(n+1)/2 + n)/Product_{k=1..n} (1-q^k)^2.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * phi^2 * n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 06 2021
a(n) = A143184(n) - A001524(n). - Vaclav Kotesovec, Jun 06 2021

a(0)=1 prepended by Alois P. Heinz, Jan 15 2021

A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 6, 12, 10, 17, 17, 22, 26, 30, 40, 40, 57, 55, 82, 74, 112, 103, 153, 140, 203, 193, 270, 262, 351, 357, 458, 478, 589, 641, 760, 846, 971, 1114, 1244, 1450, 1582, 1880, 2018, 2412, 2558, 3086, 3247, 3914, 4102, 4949

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A143184, A340622, A340623.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrt(N), x^(k^2)*(1-x^(2*k))/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

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

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Jan 14 2021

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 9, 16, 28, 49, 84, 140, 228, 361, 560, 856, 1288, 1916, 2821, 4108, 5928, 8480, 12024, 16920, 23637, 32788, 45196, 61928, 84368, 114332, 154160, 206857, 276308, 367476, 486680, 641996, 843656, 1104592, 1441168, 1873965, 2428816, 3138132, 4042408, 5192132

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A143184, A192918, A376812, A376852, A376854.

nmax = 60; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[(1+x^j)/(1-x^j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(sqrt(8*n*(log(r)^2 + polylog(2,r) - polylog(2,-r)))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r*(1+r^2) = (1-r^2) and c = 0.0643033662740307713580663125340126524175...

A132211 Coefficients of a Ramanujan q-series.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -1, 1, -1, 0, 1, -1, 1, -2, 3, -4, 4, -5, 7, -8, 8, -9, 11, -12, 12, -13, 15, -16, 16, -17, 19, -20, 19, -20, 22, -22, 21, -21, 22, -22, 20, -19, 20, -19, 16, -14, 14, -12, 8, -5, 3, 0, -5, 10, -13, 17, -24, 30, -34, 40, -48, 55, -61, 68, -77, 86, -93, 101

Offset: 0

Michael Somos, Aug 13 2007

sign

			G.f. = 1 - x - x^6 + x^7 - x^8 + x^9 - x^10 + 2*x^11 - 2*x^12 + 2*x^13 - 2*x^14 + ...

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 10

Convolution with A015128 is A143184. - Michael Somos, Dec 13 2022

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2) / QPochhammer[ x^2, x^2, k], {k, 0, Sqrt[8 n + 1]}], {x, 0, n}]]; (* Michael Somos, Nov 01 2015 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = -t * x^k / (1 - x^(2*k)) + x * O(x^n), 1), n))};

G.f.: Sum_{k>=0} (-1)^k * x^(k*(k + 1)/2) / (x^2; x^2)_n.

A347206 The number of overpartitions of n whose Frobenius symbols have only odd parts in the top row.

Original entry on oeis.org

1, 0, 2, 2, 4, 4, 8, 10, 16, 20, 30, 38, 54, 68, 94, 120, 160, 202, 266, 334, 432, 540, 688, 856, 1080, 1334, 1668, 2052, 2542, 3110, 3828, 4660, 5698, 6906, 8394, 10130, 12250, 14720, 17716, 21210, 25412, 30310, 36172, 42994, 51114, 60558, 71740, 84732, 100052

Offset: 0

Jeremy Lovejoy, Aug 23 2021

nonn

a(n) is also the excess of the number of overpartitions of n with an even number of non-overlined parts larger than the number of overlined parts over the number of overpartitions of n with an odd number of non-overlined parts larger than the number of overlined parts.

B. Kim, E. Kim, and J. Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int. J. Number Theory 17 (2021), 603-619.

Cf. A015128, A143184, A347207.

G.f.: (Product_{k>=1} 1/(1-q^k))*Sum_{n>=0} q^(n*(3*n+1)/2)*(1-q^(2*n+1)).

A347207 The number of overpartitions of n whose Frobenius symbols have only positive parts in the top row.

Original entry on oeis.org

1, 0, 2, 4, 6, 10, 16, 26, 40, 62, 92, 136, 198, 284, 404, 570, 794, 1100, 1512, 2060, 2792, 3760, 5030, 6696, 8868, 11682, 15322, 20008, 26012, 33688, 43464, 55864, 71560, 91360, 116256, 147490, 186562, 235304, 295976, 371308, 464614, 579944, 722180, 897212

Offset: 0

Jeremy Lovejoy, Aug 23 2021

nonn

a(n) is also the excess of the number of overpartitions of n with an even number of overlined parts larger than the number of non-overlined parts over the number of overpartitions of n with an odd number of overlined parts larger than the number of non-overlined parts.

B. Kim, E. Kim, and J. Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int. J. Number Theory 17 (2021), 603-619.

Cf. A015128, A143184, A347206.

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k))*(1-2*Sum_{n>=1} q^(n*(3*n-1)/2)*(1-q^n)).

A376707 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^3.

Original entry on oeis.org

1, 1, 3, 7, 13, 24, 41, 70, 114, 186, 293, 459, 703, 1067, 1593, 2359, 3447, 4998, 7175, 10222, 14445, 20277, 28263, 39156, 53922, 73843, 100587, 136331, 183890, 246909, 330094, 439453, 582738, 769782, 1013169, 1328805, 1736942, 2263018, 2939280, 3806072, 4914221

Offset: 0

Vaclav Kotesovec, Oct 02 2024

nonn

Cf. A000009, A143184, A376710.

Cf. A263719.

nmax = 40; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^j, {j, 1, k}]^3, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ r^(1/6) * (log(r)^2 + 6*polylog(2, 1-r))^(3/4) * exp(sqrt(2*(log(r)^2 + 6*polylog(2, 1-r))*n)) / (2^(11/4) * Pi^(3/2) * sqrt(1 + 2*r) * n^(5/4)), where r = 1 - A263719 = 0.3176721961719... is the real root of the equation r = (1-r)^3.

A376710 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^4.

Original entry on oeis.org

1, 1, 4, 11, 24, 49, 93, 173, 310, 549, 946, 1608, 2676, 4391, 7083, 11283, 17724, 27539, 42309, 64382, 97052, 145092, 215161, 316737, 462980, 672310, 970154, 1391667, 1984999, 2816059, 3974475, 5581789, 7802161, 10856466, 15040941, 20751416, 28515375, 39033040

Offset: 0

Vaclav Kotesovec, Oct 02 2024

nonn

Cf. A000009, A143184, A376707.

nmax = 40; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^j, {j, 1, k}]^4, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ r^(1/8) * (log(r)^2 + 8*polylog(2, 1-r)) * exp(sqrt(2*(log(r)^2 + 8*polylog(2, 1-r))*n)) / (2^(7/2) * Pi^2 * sqrt(1 + 3*r) * n^(3/2)), where r = 0.2755080409994... is the smallest real root of the equation r = (1-r)^4.

cp's OEIS Frontend

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

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A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

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A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

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A376853 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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A132211 Coefficients of a Ramanujan q-series.

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Mathematica

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A347206 The number of overpartitions of n whose Frobenius symbols have only odd parts in the top row.

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A347207 The number of overpartitions of n whose Frobenius symbols have only positive parts in the top row.

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A376707 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} 1/(1 - x^j)^3.

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A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

A376707 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^3.

A376710 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^4.