A100405 -id:A100405 - cp's OEIS frontend

A007690 Number of partitions of n in which no part occurs just once.

1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 16, 11, 22, 20, 33, 28, 51, 42, 71, 66, 100, 92, 147, 131, 199, 193, 275, 263, 385, 364, 516, 511, 694, 686, 946, 925, 1246, 1260, 1650, 1663, 2194, 2202, 2857, 2928, 3721, 3813, 4866, 4967, 6257, 6487, 8051, 8342, 10369

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

Also number of partitions of n into parts, each larger than 1, such that consecutive integers do not both appear as parts. Example: a(6)=4 because we have [6], [4,2], [3,3] and [2,2,2]. - Emeric Deutsch, Feb 16 2006
Also number of partitions of n into parts divisible by 2 or 3. - Alexander E. Holroyd (holroyd(AT)math.ubc.ca), May 28 2008
Infinite convolution product of [1,0,1,1,1,1,1] aerated n-1 times. i.e. [1,0,1,1,1,1,1] * [1,0,0,0,1,0,1] * [1,0,0,0,0,0,1] * ... . - Mats Granvik, Aug 07 2009

			a(6) = 4 because we have [3,3], [2,2,2], [2,2,1,1] and [1,1,1,1,1,1].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 6*x^8 + 5*x^9 + 9*x^10 + ...
G.f. = q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...

G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627
G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.6).
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 54, Article 300.
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
A. E. Holroyd, T. M. Liggett and D. Romik, Integrals, partitions and cellular automata, arXiv:math/0302216 [math.PR], 2003.
Eric Weisstein's World of Mathematics, Partition Function P

Cf. A000041, A055922, A055923, A114917, A114918, A183558, A183568.

Cf. A100405, A160974-A160990.

G:= mul((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x, 60): seq(coeff(Gser, x, n), n=0..54); # Emeric Deutsch, Feb 10 2006

nn=40;CoefficientList[Series[Product[1/(1-x^i)-x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Dec 02 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)
Table[Length@Select[Tally /@ IntegerPartitions@n, AllTrue[#, Last[#] > 1 &] &], {n, 0, 54}] (* Robert Price, Aug 17 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Apr 21 2004 */

G.f.: Product_{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower, Jun 23 2000
G.f.: Product_{j>=1} (1+x^(3*j)) / (1-x^(2*j)). - Jon Perry, Mar 29 2004
Euler transform of period 6 sequence [0, 1, 1, 1, 0, 1, ...]. - Michael Somos, Apr 21 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 1/6 (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137566. - Michael Somos, Jan 26 2008
From Alois P. Heinz, Oct 09 2011: (Start)
a(n) = A000041(n) - A183558(n).
a(n) = A183568(n,0) - A183568(n,1).
G.f.: Product_{j>0} (1-x^j+x^(2*j)) / (1-x^j). (End)
a(n) ~ exp(2*Pi*sqrt(n)/3)/(6*sqrt(2)*n). - Vaclav Kotesovec, Sep 23 2015
a(n) = A000009(n/3) - Sum_{k>=1} (-1)^k a(n - k*(3*k +/- 1)). - Peter J. Taylor, May 16 2019

Minor edits by Vaclav Kotesovec, Aug 23 2015

A116932 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 6, 9, 12, 14, 16, 24, 25, 32, 40, 49, 56, 73, 81, 102, 120, 142, 162, 202, 227, 270, 316, 367, 419, 506, 565, 663, 767, 879, 998, 1179, 1317, 1517, 1739, 1979, 2232, 2588, 2883, 3295, 3742, 4220, 4737, 5426, 6037, 6828, 7701, 8642, 9651, 10939

Offset: 1

Emeric Deutsch, Feb 27 2006

nonn

Also, partitions of n in which any two distinct parts differ by at least 3. Example: a(5) = 3 because we have [5], [4,1] and [1,1,1,1,1].

			a(5) = 3 because we have [5], [2,1,1,1] and [1,1,1,1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A116931, A100405.

Column k=3 of A218698. - Alois P. Heinz, Nov 04 2012

g:=sum(x^k*product(1+x^(3*j)/(1-x^j),j=1..k-1)/(1-x^k),k=1..70): gser:=series(g,x=0,62): seq(coeff(gser,x^n),n=1..58);
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-3), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 04 2012

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-3], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

G.f.: sum(x^k*product(1+x^(3j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869... - Vaclav Kotesovec, Jan 28 2022

A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 314, 417, 552, 733, 955, 1244, 1617, 2079, 2665, 3413, 4331, 5485, 6931, 8704, 10901, 13629, 16949, 21033, 26045, 32123, 39529, 48553, 59429, 72599, 88518, 107624, 130599, 158209, 191175, 230611, 277717, 333730, 400375, 479598, 573386, 684481

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

a(n) is the number of overpartitions wherein only parts that are a multiple of three may be overlined. - Alois P. Heinz, Feb 03 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020, p. 6.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000726, A015128, A100405, A266647, A266649, A266650, A285445, A285447.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2025

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(7) * exp(sqrt(7*n)*Pi/3) / (24*n).

A353400 Number of integer compositions of n with all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 2, 4, 4, 5, 11, 11, 14, 27, 29, 37, 61, 72, 97, 147, 181, 246, 368, 470, 632, 914, 1198, 1611, 2286, 3018, 4079, 5709, 7619, 10329, 14333, 19258, 26142, 36069, 48688, 66114, 90800, 122913, 167020, 228735, 310167, 421708, 576499, 782803

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(7) = 1 through a(12) = 11 compositions:
  1111111   2222       333         22222        1112222       444
            11111111   111222      1111222      2222111       3333
                       222111      2221111      11111222      111333
                       111111111   1111111111   22211111      222222
                                                11111111111   333111
                                                              11112222
                                                              22221111
                                                              111111222
                                                              111222111
                                                              222111111
                                                              111111111111

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The = 2 version is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A078012, both A353428.
The version for partitions is A100405.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A274174 counts compositions with equal parts contiguous.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(21)-a(49) from Alois P. Heinz, May 17 2022

A353401 Number of integer compositions of n with all prime run-lengths.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 3, 6, 9, 10, 18, 27, 35, 54, 83, 107, 176, 242, 354, 515, 774, 1070, 1648, 2332, 3429, 4984, 7326, 10521, 15591, 22517, 32908, 48048, 70044, 101903, 149081, 216973, 316289, 461959, 672664, 981356, 1431256, 2086901, 3041577, 4439226, 6467735

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
  0   .  11   111   22   11111   33     11122     44       333
                                 222    22111     1133     11133
                                 1122   1111111   3311     33111
                                 2211             11222    111222
                                                  22211    222111
                                                  112211   1111122
                                                           1112211
                                                           1122111
                                                           2211111

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of runs equal to 2 is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A023360, both A353429.
The version for partitions is A055923.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
The > 2 version is A353400, partitions A100405.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.
A005811 counts runs in binary expansion.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A167606 counts compositions with adjacent parts coprime.
A329738 counts uniform compositions, partitions A047966.
Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
     `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Length/@Split[#],_?(!PrimeQ[#]&)]&]],{n,0,15}]

a(21)-a(45) from Alois P. Heinz, May 18 2022

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698

Offset: 0

Vaclav Kotesovec, Nov 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A266647 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 9, 15, 27, 46, 75, 118, 187, 285, 429, 639, 935, 1354, 1945, 2758, 3878, 5417, 7493, 10300, 14070, 19087, 25741, 34542, 46081, 61185, 80869, 106391, 139368, 181867, 236357, 306060, 394939, 507860, 650946, 831792, 1059600, 1345920, 1704880, 2153682

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

Convolution of A264905 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A100405, A264905, A266648, A266649, A266650.

nmax = 50; CoefficientList[Series[Product[(1+x^k+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (12*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A266650 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 53, 82, 123, 181, 263, 379, 537, 754, 1047, 1444, 1972, 2675, 3601, 4820, 6408, 8473, 11141, 14580, 18985, 24611, 31765, 40839, 52294, 66719, 84819, 107474, 135731, 170892, 214518, 268524, 335190, 417308, 518212, 641948, 793324, 978157

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

Convolution of A266686 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266649, A266686.

nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A266649 Expansion of Product_{k>=1} 1 - x^(3*k)/(1-x^k).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, -2, -1, -2, -1, -1, 1, -1, 3, 2, 5, 4, 8, 7, 11, 8, 12, 11, 13, 10, 9, 9, 7, 3, -2, -5, -13, -16, -25, -28, -48, -44, -66, -60, -82, -82, -104, -95, -120, -103, -131, -107, -133, -98, -124, -85, -94, -42, -51, 9, 7, 83, 100, 181, 208, 298

Offset: 0

Vaclav Kotesovec, Jan 02 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266650.

nmax = 100; CoefficientList[Series[Product[1-x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k - x^(3*k)) / (1 - x^k).

A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 4, 0, 0, 8, 3, 0, 10, 4, 4, 15, 4, 8, 24, 7, 8, 42, 16, 10, 59, 31, 27, 87, 37, 52, 149, 62, 66, 233, 121, 111, 342, 207, 204, 531, 308, 351, 864, 487, 536, 1373, 864, 865, 2057, 1440, 1509, 3232

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) compositions for selected n:
  n=16:   n=18:     n=20:    n=21:      n=24:
----------------------------------------------------
  (4444)  (666)     (5555)   (777)      (888)
          (333333)  (44444)  (333444)   (6666)
                             (444333)   (333555)
                             (3333333)  (444444)
                                        (555333)
                                        (3333444)
                                        (4443333)
                                        (33333333)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Allowing any multiplicities gives A078012, partitions A008483.
The version for no (instead of all) parts or run-lengths > 2 is A137200.
Allowing any parts gives A353400, partitions A100405.
The version for partitions is A353501, ranked by A353502.
The version for > 1 instead of > 2 is A353508, partitions A339222.
A003242 counts anti-run compositions, ranked by A333489.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A128695 counts compositions with no run-lengths > 2.
A261983 counts non-anti-run compositions.
A335464 counts compositions with a run-length > 2.
Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(26)-a(66) from Alois P. Heinz, May 17 2022

cp's OEIS Frontend

A007690 Number of partitions of n in which no part occurs just once.

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A116932 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times.

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A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

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A353400 Number of integer compositions of n with all run-lengths > 2.

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A353401 Number of integer compositions of n with all prime run-lengths.

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A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

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A266647 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)) / (1 - x^k).

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