cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A221584 A sum over partitions (q=20), see first comment.

Original entry on oeis.org

1, 19, 399, 7980, 159980, 3199581, 63999600, 1279991601, 25599991620, 511999832020, 10239999832020, 204799996632420, 4095999996640419, 81919999932640800, 1638399999932648400, 32767999998652808799, 655359999998652816380, 13107199999973052976380
Offset: 0

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Author

Joerg Arndt, Jan 20 2013

Keywords

Comments

Set q=20 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Links

Programs

  • PARI
    N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-20*x^n)  );
v=Vec(gf)

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

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

Examples

			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).

References

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

Links

Crossrefs

Cf. A001970.
Cf. A006951, A006952, A049314, A049315, A049316.

Programs

  • GAP
    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
  • GAP
    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
  • Mathematica
    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. *)

Formula

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

Extensions

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

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

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Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

Examples

			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,  ...

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Formula

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).
Previous Showing 21-23 of 23 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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