A006951 -id:A006951 - cp's OEIS frontend

A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

1, 1, 4, 10, 29, 72, 200, 510, 1364, 3546, 9348, 24400, 64090, 167562, 439200, 1149360, 3010349, 7879832, 20633304, 54014950, 141422328, 370239300, 969323000, 2537696160, 6643839400, 17393731933, 45537549048, 119218684970, 312119004990, 817137724392, 2139295489200, 5600747143950

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Euler transform of A032198.

Alois P. Heinz, Table of n, a(n) for n = 0..2392
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.

Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163, A386706, A387017.

Convolution inverse of A329157.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(a(j)*b(n-j$2), j=0..n-1))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 25 2025

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.
a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019
a(2^k) = A002878(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 on page 11 of Kassel-Reutenauer paper. - Michael De Vlieger, Jul 28 2025

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal sums is A318683.
The version with strictly decreasing sums is A318684.
The version with weakly decreasing sums is A319794.
The version with different sums is A336132.
Starting with a composition gives A304961.
Starting with a non-strict partition gives A336134.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 15 splittings:
  (1)  (2)      (3)          (4)              (5)
       (1,1)    (2,1)        (2,2)            (3,2)
       (1),(1)  (1,1,1)      (3,1)            (4,1)
                (1),(1,1)    (2,1,1)          (2,2,1)
                (1),(1),(1)  (2),(2)          (3,1,1)
                             (1,1,1,1)        (2,1,1,1)
                             (2),(1,1)        (2),(2,1)
                             (1),(1,1,1)      (1,1,1,1,1)
                             (1,1),(1,1)      (2),(1,1,1)
                             (1),(1),(1,1)    (1),(1,1,1,1)
                             (1),(1),(1),(1)  (1,1),(1,1,1)
                                              (1),(1),(1,1,1)
                                              (1),(1,1),(1,1)
                                              (1),(1),(1),(1,1)
                                              (1),(1),(1),(1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..60

The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

Original entry on oeis.org

1, 3, 9, 24, 60, 141, 324, 717, 1560, 3330, 7020, 14622, 30225, 61998, 126522, 257007, 520326, 1050396, 2116116, 4255584, 8547330, 17149350, 34382295, 68889840, 137969466, 276220962, 552865365, 1106356314, 2213644548, 4428657402, 8859340926, 17721640698

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A006951, A070933, A261584, A264686.

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

a(n) ~ c * 2^n, where c = A079555 / A048651 = Product_{k>=1} (2^k+1)/(2^k-1) = 8.25598793577825006554414084943227312652...

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 6, 10, 12, 18, 30, 40, 48, 74, 92, 142, 172, 242, 294, 412, 490, 722, 854, 1164, 1396, 1880, 2260, 2976, 3748, 4764, 5792, 7472, 9082, 11488, 14012, 17522, 21830, 26896, 32820, 40536, 49488, 60636, 73626, 89962, 108854, 134240, 160952, 195858

Offset: 0

Gus Wiseman, Jan 20 2019

nonn

			The a(9) = 18 plane partitions:
  81   72   621   54   531   432
.
  8   7   61   62   5   51   53   42   43
  1   2   2    1    4   3    1    3    2
.
  6   5   4
  2   3   3
  1   1   2

Cf. A000219, A000837, A003293, A006951, A026007, A100883, A117433 (strict plane partitions), A300275 (plane partitions with relatively prime parts), A303546, A320802, A323584, A323585.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]}],{n,30}]

Moebius transform of A117433.

A124679 a(n) = number of conjugacy classes in PSL_3(prime(n)).

Original entry on oeis.org

6, 12, 30, 22, 132, 64, 306, 130, 552, 870, 334, 472, 1722, 634, 2256, 2862, 3540, 1264, 1522, 5112, 1804, 2110, 6972, 8010, 3172, 10302, 3574, 11556, 4000, 12882, 5422, 17292, 18906, 6490, 22350, 7654, 8272, 8914, 28056, 30102, 32220, 10984

Offset: 1

N. J. A. Sloane, Dec 25 2006

Jason Kimberley, Table of n, a(n) for n = 1..54
Jason Kimberley, Incomplete table of n, a(n) for n = 1..67

Cf. A000702, A006951, A124678.

A124679 := func< n | NumberOfClasses(PSL(3,NthPrime(n))) >;

a(7) to a(14) from Klaus Brockhaus, Dec 26 2006
a(15)..a(54) appended, from running MAGMA for 7 processor days at U. Newcastle, by Jason Kimberley, Feb 25 2011.
a(65)=32764 added to a124679.txt by Jason Kimberley, Mar 28 2011.

A003606 a(n) = number of types of conjugacy classes in GL(n,q) (this is independent of q).

Original entry on oeis.org

1, 4, 8, 22, 42, 103, 199, 441, 859, 1784, 3435, 6882, 13067, 25366, 47623, 90312, 167344, 311603, 570496, 1045896, 1893886, 3426466, 6140824, 10984249, 19499214, 34526844, 60758733, 106613119, 186099976, 323883380, 561141244, 969308408

Offset: 1

N. J. A. Sloane, Mira Bernstein

			a(2) = 4 as there are four types of conjugacy classes of 2 X 2 matrices over GF(q):
* the scalar matrices (diagonal matrix with both entries the same)
* the direct sum of two scalars (diagonal matrix with both entries different)
* the non-diagonalizable Jordan block (upper triangular matrix with the same entry along the diagonal and a 1 in the superdiagonal)
* companion matrices of irreducible quadratics over GF(q)
This example can be found in Green's paper (in the references).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..500
J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955), 402-447.
N. J. A. Sloane, Transforms.
R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc., 71 (1951), 274-282.

Cf. A001970.

Cf. A006951, A006952, A049314, A049315, A049316.

a := function(n) local k,sum; sum := 0; for k in [0..n-1] do sum := sum + a(k)*g(n-k); od; return sum/n; end;
g := function(n) local i,j,sum; for i in DivisorsInt(n) do for j in DivisorsInt(n/i) do sum := sum + NrPartitions(i)*i*j; od; od; return sum; end;;
# This code is significantly faster if you store previously computed values of a(n) and g(n).
# Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003

a := function(n) if( n = 0) then return 1; else return Sum([0..n], i -> t(i) * Sum(DivisorsInt(n-i), d -> d * NrPartitions(d) * Sigma(n/d)) )/n; fi; end;; # Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006

m = 32; f[x_] = Product[1/(1-x^k), {k, 1, m}]; gf[x_] = Product[f[x^k]^PartitionsP[k], {k, 1, m}]; Drop[ CoefficientList[ Series[gf[x], {x, 0, m}], x], 1] (* Jean-François Alcover, Aug 01 2011, after g.f. *)

G.f.: Product_{k >= 1} f(x^k)^p_k, where f(x) = Product_{k >= 0} 1/(1-x^k) = Sum_{k >= 0} p_k*x^k and p_k is the number of partitions of k (A000041).
Recurrence relation: a(n+1) = (1/(n+1)) * Sum_{k=0..n} a(k)*g(n-k+1) where g(n) = Sum_{i*j | n} p(i)*i*j, with the sum over all ordered pairs (i, j) such that their products divide n and p(i) is the number of partitions of i. Also a(0)=1. - Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003
Euler transform of A047968(n). - Vladeta Jovovic, Jun 23 2004
Recurrence relation: a(0)=1, a(n+1) = (1/(n+1)) * Sum_{k=0..n} a(k)*g(n-k+1) where g(n) = Sum_{d | n} d * A000041(d) * A000203(n/d). - Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006

More terms from Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003

A070731 Size of largest conjugacy class in the group GL(n,2).

Original entry on oeis.org

1, 3, 56, 3360, 833280, 959938560, 3901190307840, 63667425823948800, 4759267415191820697600, 1246395024829755538853068800

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 15 2002

nonn

V. Jovovic, Cycle indices of general linear group GL(n,2)

Cf. A006951, A002884.

More terms from Vladeta Jovovic, Jun 03 2002

A124681 a(n) = number of conjugacy classes in PSL_4(prime(n)).

Original entry on oeis.org

14, 29, 49, 217, 757, 613, 1327, 3661, 6409, 6349, 15457, 13057, 17707, 40789, 53137, 37993, 104581, 57757, 152797

Offset: 1

N. J. A. Sloane, Dec 25 2006

Cf. A000702, A006951, A124679, A124678.

A124681 := func< n | NumberOfClasses(PSL(4,NthPrime(n))) >;

a(5) and a(6) from Klaus Brockhaus, Dec 26 2006 and  Oct 09 2010
a(7)..a(14) appended, from running MAGMA for 32 processor hours at U. Newcastle, by Jason Kimberley, Feb 09 2011.
a(15)-a(19) from Robin Visser, Oct 01 2023

A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 1, 3, 8, 6, 0, 1, 1, 4, 15, 24, 14, 0, 0, 1, 5, 24, 60, 78, 27, 0, 1, 1, 6, 35, 120, 252, 232, 60, 0, 0, 1, 7, 48, 210, 620, 1005, 720, 117, 0, 0, 1, 8, 63, 336, 1290, 3096, 4080, 2152, 246, 0, 0, 1, 9, 80, 504, 2394, 7735, 15600, 16305, 6528, 490, 0, 0

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
  -1,  0,   1,    2,     3,     4,  ...
  -1,  0,   3,    8,    15,    24,  ...
   0,  0,   6,   24,    60,   120,  ...
   0,  0,  14,   78,   252,   620,  ...
   1,  0,  27,  232,  1005,  3096,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..20 give A010815, A000007, A006951, A006952, A049314, A049315, A221578, A049316, A182603, A182604, A221579, A182605, A221580, A182606, A221581, A221582, A182607, A182608, A221583, A182609, A221584.

Main diagonal gives A319754.

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

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

G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).

cp's OEIS Frontend

A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A124679 a(n) = number of conjugacy classes in PSL_3(prime(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Extensions

A003606 a(n) = number of types of conjugacy classes in GL(n,q) (this is independent of q).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

GAP

GAP

Mathematica

Formula

Extensions

A070731 Size of largest conjugacy class in the group GL(n,2).

Views

Author

Keywords

Links

Crossrefs

Extensions