A218698 -id:A218698 - cp's OEIS frontend

A006951 Number of conjugacy classes in GL(n,2).

1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053, 8088, 16284, 32559, 65330, 130626, 261726, 523374, 1047690, 2095314, 4192479, 8384808, 16773552, 33546736, 67101273, 134202258, 268420086, 536839446, 1073710914, 2147420250, 4294904430, 8589807438

Offset: 0

N. J. A. Sloane

nonn

Unlabeled permutations of sets. - Christian G. Bower, Jan 29 2004
From Joerg Arndt, Jan 02 2013: (Start)
Set q=2 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L], see the Macdonald reference.
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power are:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.
(End)
From Gus Wiseman, Jan 21 2019: (Start)
Also the number of ways to split an integer partition of n into consecutive constant subsequences. For example, the a(5) = 27 ways (subsequences shown as rows) are:
  5   11111
.
  4   3   3    22   2     1111   1      111   11
  1   2   11   1    111   1      1111   11    111
.
  3   2   2    2    111   1     1     11   11   1
  1   2   11   1    1     111   1     11   1    11
  1   1   1    11   1     1     111   1    11   11
.
  2   11   1    1    1
  1   1    11   1    1
  1   1    1    11   1
  1   1    1    1    11
.
  1
  1
  1
  1
  1
(End)

			For the 5 partitions of 4 (namely [1^4]; [2,1^2]; [2^2]; [3,1]; [4]) we have
(f(m) = 2^(m-1)*(2-1) = 2^(m-1) and)
f([1^4]) = 2^3 = 8,
f([2,1^2]) = 1*2^1 = 2,
f([2^2]) = 2^1 = 2,
f([3,1]) = 1*1 = 1,
f([4]) = 1,
the sum is 8+2+2+1+1 = 14 = a(4).
- _Joerg Arndt_, Jan 02 2013

W. D. Smith, personal communication.
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..1000
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161
I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).
N. J. A. Sloane, Transforms

Cf. A006952, A049314, A049315, A049316, A070933, A264685, A264687.
Column k=0 of A218698. - Alois P. Heinz, Nov 04 2012
Cf. A100471, A100883, A279784, A279786, A323433, A323582, A323583.

/* The program does not work for n>19: */
[1] cat [NumberOfClasses(GL(n,2)): n in [1..19]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi Jan 24 2013

with(numtheory):
b:= n-> add(phi(d)*2^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012

b[n_] := Sum[EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Sum[2^(Length[ptn]-Length[Split[ptn]]),{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 21 2019 *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-2*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */

G.f.: Product_{n>=1} (1-x^n)/(1-2*x^n). - Joerg Arndt, Jan 02 2013
The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in Product_{k>=1} (1-t^k)/(1-q*t^k). - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
Euler transform of A008965. - Christian G. Bower, Jan 29 2004
a(n) ~ 2^n - (1+sqrt(2) + (-1)^n*(1-sqrt(2))) * 2^(n/2-1). - Vaclav Kotesovec, Nov 21 2015
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(2^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

More terms from Christian G. Bower, Jan 29 2004

A116931 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least twice.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 13, 15, 22, 24, 37, 40, 57, 64, 89, 98, 135, 149, 199, 224, 292, 325, 424, 472, 601, 676, 850, 950, 1191, 1329, 1643, 1845, 2258, 2524, 3082, 3442, 4158, 4659, 5591, 6246, 7472, 8338, 9903, 11072, 13077, 14586, 17184, 19150, 22431, 25019

Offset: 1

Emeric Deutsch, Feb 27 2006

nonn

Also, partitions of n in which any two distinct parts differ by at least 2. Example: a(5) = 4 because we have [5], [4,1], [3,1,1] and [1,1,1,1,1].

			a(5) = 4 because we have [5], [3,1,1], [2,1,1,1] and [1,1,1,1,1].
q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 13*q^8 + 15*q^9 + ...
There are a(9) = 15 partitions of 9 where distinct parts differ by at least 2:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 3 1 1 1 1 1 1 ]
03:  [ 3 3 1 1 1 ]
04:  [ 3 3 3 ]
05:  [ 4 1 1 1 1 1 ]
06:  [ 4 4 1 ]
07:  [ 5 1 1 1 1 ]
08:  [ 5 2 2 ]
09:  [ 5 3 1 ]
10:  [ 6 1 1 1 ]
11:  [ 6 3 ]
12:  [ 7 1 1 ]
13:  [ 7 2 ]
14:  [ 8 1 ]
15:  [ 9 ]
- _Joerg Arndt_, Jun 09 2013

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 52, Article 298.

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000 (terms 1..7500 from Alois P. Heinz)
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A116932, A007690.
Column k=2 of A218698. - Alois P. Heinz, Nov 04 2012
Column k=0 of A268193. - Alois P. Heinz, Feb 13 2016

g:=sum(x^k*product(1+x^(2*j)/(1-x^j),j=1..k-1)/(1-x^k),k=1..70): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..54);
# 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-2), 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-2], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k) * prod( j=1, k-1, 1 + x^(2*j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))} /* Michael Somos, Jan 26 2008 */

G.f.: sum(x^k*product(1+x^(2j)/(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)) ~ 2*Pi*sqrt(n)/3. - Vaclav Kotesovec, Jan 28 2022

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

A218699 Number of partitions of n in which any two distinct parts differ by at least 4.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 4, 8, 8, 12, 12, 19, 18, 24, 26, 36, 36, 48, 50, 70, 71, 92, 96, 129, 133, 168, 177, 225, 233, 294, 307, 382, 401, 488, 518, 635, 668, 803, 855, 1027, 1089, 1298, 1381, 1638, 1745, 2047, 2184, 2569, 2734, 3181, 3404, 3953, 4213, 4863, 5203

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 4 times.

			a(5) = 2: [1,1,1,1,1], [5].
a(6) = 5: [1,1,1,1,1,1], [2,2,2], [3,3], [1,5], [6].
a(7) = 4: [1,1,1,1,1,1,1], [1,1,5], [1,6], [7].
a(8) = 8: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,1,5], [1,1,6], [2,6], [1,7], [8].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=4 of A218698.

Cf. A160974.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-4), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i j, i - k, k], {j, 1, n/i}]]];
a[n_] := b[n, n, 4];
a /@ Range[0, 70] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(4*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-4*x)) dx = -0.9030055506558938921393786530232872470622617736... - Vaclav Kotesovec, Jan 28 2022

A218700 Number of partitions of n in which any two distinct parts differ by at least 5.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 6, 7, 9, 10, 15, 15, 19, 23, 26, 28, 36, 37, 48, 52, 62, 67, 85, 93, 110, 122, 144, 157, 194, 205, 241, 265, 304, 338, 391, 422, 483, 533, 607, 661, 760, 822, 933, 1032, 1151, 1260, 1432, 1554, 1751, 1920, 2137, 2333, 2621, 2848, 3176

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 5 times.

			a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
a(7) = 3: [1,1,1,1,1,1,1], [1,6], [7].
a(8) = 6: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,6], [1,7], [8].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=5 of A218698.

Cf. A160975.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-5), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i j, i - k, k], {j, 1, n/i}]]];
a[n_] := b[n, n, 5];
a /@ Range[0, 70] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(5*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-5*x)) dx = -0.9908078441778424729564846063206238729218368028... - Vaclav Kotesovec, Jan 28 2022

A218701 Number of partitions of n in which any two distinct parts differ by at least 6.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 5, 8, 7, 14, 11, 16, 19, 23, 22, 32, 29, 38, 40, 48, 48, 67, 63, 81, 85, 106, 106, 141, 138, 174, 180, 219, 224, 284, 282, 342, 356, 422, 431, 530, 532, 631, 660, 765, 789, 948, 965, 1123, 1184, 1356, 1408, 1658, 1703, 1967, 2076

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 6 times.

			a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
a(7) = 2: [1,1,1,1,1,1,1], [7].
a(8) = 5: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,7], [8].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=6 of A218698.

Cf. A160976.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-6), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(6*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-6*x)) dx = -1.0551351195231385243969621008395374852115209081... - Vaclav Kotesovec, Jan 28 2022

A218702 Number of partitions of n in which any two distinct parts differ by at least 7.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 4, 6, 6, 11, 10, 13, 16, 19, 20, 25, 27, 33, 34, 39, 41, 51, 52, 61, 65, 80, 82, 99, 104, 126, 133, 156, 168, 199, 209, 243, 261, 302, 320, 372, 392, 447, 479, 537, 572, 650, 693, 770, 829, 920, 982, 1096, 1169, 1306, 1396, 1541

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 7 times.

			a(8) = 4: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8].
a(9) = 4: [1,1,1,1,1,1,1,1,1], [3,3,3], [1,8], [9].
a(10) = 6: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,1,8], [1,9], [10].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=7 of A218698.

Cf. A160977.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-7), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(7*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-7*x)) dx = -1.104868234083422137620242346741601264555358762... - Vaclav Kotesovec, Jan 28 2022

A218703 Number of partitions of n in which any two distinct parts differ by at least 8.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 4, 10, 7, 12, 13, 17, 16, 23, 21, 30, 30, 34, 35, 47, 43, 51, 52, 66, 63, 81, 77, 100, 99, 120, 121, 156, 150, 185, 189, 234, 230, 283, 281, 343, 350, 409, 414, 503, 497, 587, 600, 695, 703, 824, 830, 967, 988, 1122, 1148, 1333

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 8 times.

			a(9) = 3: [1,1,1,1,1,1,1,1,1], [3,3,3], [9].
a(10) = 5: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,9], [10].
a(11) = 4: [1,1,1,1,1,1,1,1,1,1,1], [1,1,9], [1,10], [11].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=8 of A218698.

Cf. A160978.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-8), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(8*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-8*x)) dx = -1.1447921975208768146551512630331558734964408879... - Vaclav Kotesovec, Jan 28 2022

A218704 Number of partitions of n in which any two distinct parts differ by at least 9.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 3, 8, 6, 9, 12, 14, 14, 20, 19, 24, 28, 29, 31, 41, 40, 44, 49, 54, 54, 69, 64, 77, 82, 94, 98, 119, 118, 139, 149, 173, 178, 215, 217, 253, 274, 306, 320, 375, 385, 440, 469, 521, 545, 626, 647, 718, 769, 838, 881, 994, 1026

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 9 times.

			a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
a(11) = 3: [1,1,1,1,1,1,1,1,1,1,1], [1,10], [11].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=9 of A218698.

Cf. A160979.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-9), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(9*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-9*x)) dx = -1.1777463281753839250026821894706693162991897931... - Vaclav Kotesovec, Jan 28 2022

A218705 Number of partitions of n in which any two distinct parts differ by at least 10.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 7, 4, 8, 9, 13, 11, 18, 16, 23, 22, 27, 27, 36, 35, 41, 42, 51, 48, 61, 57, 69, 65, 80, 81, 98, 93, 115, 112, 144, 136, 170, 164, 202, 204, 244, 242, 296, 290, 353, 350, 415, 412, 493, 494, 576, 580, 671, 673, 794, 786, 903

Offset: 0

Alois P. Heinz, Nov 04 2012

nonn

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 10 times.

			a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
a(11) = 2: [1,1,1,1,1,1,1,1,1,1,1], [11].
a(12) = 7: [1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [1,11], [12].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Column k=10 of A218698.

Cf. A160980.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-10), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(10*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-10*x)) dx = -1.2055372531240537414216314471404302128615809819... - Vaclav Kotesovec, Jan 28 2022

cp's OEIS Frontend

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