A047968 -id:A047968 - cp's OEIS frontend

A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 63, 82, 109, 140, 183, 233, 298, 376, 479, 598, 753, 938, 1171, 1449, 1797, 2210, 2726, 3342, 4095, 4990, 6088, 7388, 8968, 10843, 13099, 15770, 18975, 22756, 27276, 32603, 38925, 46353, 55158, 65479, 77656, 91904, 108645

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Cf. A002865, A047968, A063834, A093637, A295924, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=(1-nn)/(1-x)+Sum[Product[1/(1-x^(d k+1)),{k,0,nn}],{d,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).

A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).

Original entry on oeis.org

1, 4, 7, 17, 25, 58, 87, 177, 289, 528, 860, 1550, 2486, 4257, 6910, 11474, 18335, 29941, 47331, 75819, 118887, 187338, 290784, 452904, 696058, 1071234, 1632947, 2487504, 3759613, 5676424, 8512310, 12744903, 18975839, 28194293, 41691157, 61516394, 90379785

Offset: 1

Ilya Gutkovskiy, Jun 20 2018

nonn

Inverse Moebius transform of A000219.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Plane Partition

Cf. A000219, A047966, A047968, A300275, A302550.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=numtheory[divisors](n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018

nmax = 37; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^k)^k , {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 37}]
b[0] = 1; b[n_] := b[n] = Sum[b[n - j] DivisorSigma[2, j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 37}]

G.f.: Sum_{k>=1} A000219(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} A000219(d).

A323764 Dirichlet self-convolution of the integer partition numbers A000041.

Original entry on oeis.org

1, 1, 4, 6, 14, 14, 34, 30, 64, 69, 112, 112, 228, 202, 330, 394, 575, 594, 956, 980, 1492, 1674, 2228, 2510, 3700, 3965, 5276, 6200, 8126, 9130, 12318, 13684, 17842, 20622, 25808, 29976, 38377, 43274, 53990, 62976, 77912, 89166, 110656, 126522, 154918, 179744

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of multiset partitions of constant multiset partitions of integer partitions of n.

			The a(4) = 14 multiset partitions of constant multiset partitions:
  ((1111))              ((22))      ((4))  ((31))  ((211))
  ((11)(11))            ((2)(2))
  ((11))((11))          ((2))((2))
  ((1)(1)(1)(1))
  ((1))((1)(1)(1))
  ((1)(1))((1)(1))
  ((1))((1))((1)(1))
  ((1))((1))((1))((1))

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

Cf. A000041, A000837, A001970, A002033, A003238, A034729, A047968, A279787, A305551, A306017, A319056, A323774.

Join[{1},Table[Sum[PartitionsP[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

A329436 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 5, 6, 8, 7, 13, 10, 16, 18, 22, 21, 34, 29, 44, 45, 56, 56, 82, 78, 100, 109, 136, 137, 185, 181, 231, 247, 295, 317, 399, 404, 490, 533, 638, 669, 817, 853, 1020, 1108, 1276, 1371, 1638, 1728, 2017, 2186, 2519, 2702, 3153, 3371, 3885

Offset: 1

Ilya Gutkovskiy, Nov 13 2019

nonn

Inverse Moebius transform of A025147.

Number of uniform (constant multiplicity) partitions of n not containing 1, ranked by the odd terms of A072774. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:
  (2)  (3)  (4)    (5)    (6)      (7)    (8)        (9)      (10)
            (2,2)  (3,2)  (3,3)    (4,3)  (4,4)      (5,4)    (5,5)
                          (4,2)    (5,2)  (5,3)      (6,3)    (6,4)
                          (2,2,2)         (6,2)      (7,2)    (7,3)
                                          (2,2,2,2)  (3,3,3)  (8,2)
                                                     (4,3,2)  (5,3,2)
                                                              (3,3,2,2)
                                                              (2,2,2,2,2)
(End)

The strict case is A025147.
The version allowing 1 is A047966.
The version requiring 1 is A097986.
Cf. A023645, A047968, A072774, A096765, A329435, A329436, A367586.

nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&SameQ@@Length/@Split[#]&]], {n,0,30}] (* Gus Wiseman, Dec 01 2023 *)

G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A025147(d).

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

A055893 Inverse Moebius transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 1, 4, 4, 3, 1, 4, 1, 3, 4, 3, 2, 1, 2, 1, 5, 5, 8, 3, 3, 1, 4, 1, 4, 8, 6, 5, 4, 2, 1, 3, 1, 6, 8, 11, 8, 7, 3, 3, 1, 4, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 2, 1, 7, 13, 19, 13, 17, 7, 8, 4, 3, 1, 6, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 2, 1, 8, 16, 26

Offset: 1

Christian G. Bower, Jun 09 2000

			1; 1,2; 1,1,2; 1,3,1,3; 1,2,2,1,2; ...

N. J. A. Sloane, Transforms

Row sums give A047968. Cf. A055892.

A063835 Three times partitioned numbers: the number of ways a number can be partitioned in (not necessarily different) parts and each part again so partitioned a second and a third time.

Original entry on oeis.org

1, 5, 14, 51, 125, 429, 1039, 3258, 8254, 23554, 58934, 168803, 412177, 1114550, 2795446, 7345875, 18035424, 46875324, 114272057, 291692396, 709742614, 1774402071, 4290848175, 10672950659, 25572179792, 62670553073, 149978278320

Offset: 1

Wouter Meeussen, Aug 21 2001

Vaclav Kotesovec, Table of n, a(n) for n = 1..2000

Cf. A063834.

Table[Plus@@((Apply[Plus, #/. i_Integer-> PartitionsP[i], {1}]/. f->Times)& /@ Flatten[Flatten[Outer[f, Sequence@@(Partitions/@#), 1]]&/@Partitions[w]]), {w, 16}]
nmax = 40; A047968 = Table[Sum[PartitionsP[d], {d, Divisors[n]}], {n, 1, nmax}]; conv = Table[Sum[A047968[[j]]*PartitionsP[m - j], {j, 1, m}], {m, 1, nmax}]; A063835 = Rest[CoefficientList[Series[Product[1/(1 - conv[[k]]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 27 2016 *)

G.f.: 1/Product(1-b(n)*x^n, n=1..infinity), where b(n) is sum of number of partitions of parts in all partitions of n; b() is convolution of A047968() and A000041(). - Vladeta Jovovic, Nov 22 2005
From Vaclav Kotesovec, Mar 28 2016: (Start)
a(n) ~ c * 21^(n/4), where
c = 31506.382471540934704971753670563958673161001663... if mod(n,4) = 0
c = 31502.248225846169487427060315658509213347537914... if mod(n,4) = 1
c = 31506.175349116205868096360427802563935891182649... if mod(n,4) = 2
c = 31502.232274793501377850265964413938565498517297... if mod(n,4) = 3
(End)

More terms from Vladeta Jovovic, Nov 22 2005

A137587 Triangle read by rows: A051731 * A026794.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 6, 1, 0, 0, 1, 11, 3, 2, 0, 0, 1, 12, 2, 1, 0, 0, 0, 1, 20, 6, 1, 2, 0, 0, 0, 1, 25, 4, 3, 1, 0, 0, 0, 0, 1, 37, 9, 2, 1, 2, 0, 0, 0, 0, 1, 43, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 70, 16, 6, 3, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 27 2008

That is, regard A051731 and A026794 as lower triangular square matrices and multiply them, then take the lower triangle of the product,
Left column = A083710 starting (1, 2, 3, 5, 6, 11, 12, ...).
Row sums = A047968.

			First few rows of the triangle:
   1;
   2, 1;
   3, 0, 1;
   5, 2, 0, 1;
   6, 1, 0, 0, 1;
  11, 3, 2, 0, 0, 1;
  12, 2, 1, 0, 0, 0, 1;
  20, 6, 1, 2, 0, 0, 0, 1;
  25, 4, 3, 1, 0, 0, 0, 0, 1;
  ...

Cf. A026794, A051731, A083710, A047968.

Inverse mobius transform of the partition triangle, A026794.

Typo in 9th row corrected by M. F. Hasler, Jun 08 2009

A301763 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 5, 8, 13, 14, 5, 32, 7, 20, 64, 26, 6, 92, 7, 126, 199, 22, 5, 352, 252, 41, 581, 394, 7, 1832, 9, 292, 2119, 31, 3216, 4946, 10, 40, 8413, 7708, 9, 20656, 9, 2324, 53546, 24, 5, 70040, 16395, 59361, 131204, 9503, 7, 266780, 178180, 82086

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 8 rooted twice-partitions: (5), (11111), (2)(2), (2)(11), (11)(2), (11)(11), (1)(1)(1), ()()()()()().
The a(15) = 20 rooted twice-partitions:
()()()()()()()()()()()()()(),
(1)(1)(1)(1)(1)(1)(1), (111111)(111111), (1111111111111),
(111111)(222), (222)(111111), (222)(222),
(111111)(33), (222)(33), (33)(111111), (33)(222), (33)(33),
(111111)(6), (222)(6), (33)(6), (6)(111111), (6)(222), (6)(33), (6)(6),
(13).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A002865, A047968, A063834, A093637, A127524, A295924, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==n-1, 1, numdiv((n-1)/d-1)^d))) \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

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A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).

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Maple

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A323764 Dirichlet self-convolution of the integer partition numbers A000041.

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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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A329436 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

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

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A055893 Inverse Moebius transform of partition triangle A008284.

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A063835 Three times partitioned numbers: the number of ways a number can be partitioned in (not necessarily different) parts and each part again so partitioned a second and a third time.

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