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.

Showing 1-3 of 3 results.

A128628 An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, 1, 2, 2, 3, 3, 4, 5, 1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 5, 6, 7, 1, 2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 4, 5, 6, 7, 8, 1, 2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 3, 3, 4, 4, 5, 6, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9
Offset: 1

Views

Author

Alford Arnold, Mar 27 2007, Aug 01 2007

Keywords

Comments

The next level gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent.
Sequence A036043 counts the parts of numeric partitions and contains the same values on each row as the current sequence. When a node generates two branches the first branch can be mapped to cyclic partitions; all other branches map to matching partitions.
Appears to be the triangle in which the n-th row contains the number of parts of each partition of n, where the partitions are ordered as in A080577. - Jason Kimberley, May 12 2010

Examples

			The values at level three are 1, 2, and 3.
The 1 generates 1 and 2; the 2 generates 2 and 3; the 3 only generates 4.
The array begins
1
1 2
1 2 3
1 2 2 3 4
1 2 2 3 3 4 5
1 2 2 3 2 3 4 3 4 5 6

Links

Crossrefs

Cf. A000041, A060850, A128629.
Cf. A006128 (row sums), A036043.
Cf. A177740.
Cf. A308355 (limiting row sequence).

Programs

  • Mathematica
    Flatten[Table[Length /@ IntegerPartitions[n], {n, 9}]] (* T. D. Noe, Feb 27 2014 *)

A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds.

Original entry on oeis.org

22, 185, 810, 2580, 6765, 15525, 32305, 62337, 113265, 195910, 325193, 521235, 810654, 1228080, 1817910, 2636326, 3753600, 5256711, 7252300, 9869990, 13266099, 17627775, 23177583, 30178575, 38939875, 49822812, 63247635
Offset: 1

Views

Author

Alford Arnold, Apr 03 2007

Keywords

Comments

A slightly different method of calculating this sequence is described in A128627.

Examples

			From A128629 we can construct the table below:
      Deg    #  Associated sequence
  -------  ---  -------------------
        8    1    1  1   2   3    4
       44    2    3  1   3   6   10
       53   11    4  1   4   9   16
       62   11    4  1   4   9   16
       71   11    4  1   4   9   16
      332   12    6  1   6  18   40
      422   12    6  1   6  18   40
      431  111    8  1   8  27   64
      521  111    8  1   8  27   64
      611   12    6  1   6  18   40
     2222    4    7  1   5  15   35
     3221  112   12  1  12  54  160
     3311   22    9  1   9  36  100
     4211  112   12  1  12  54  160
     5111   13   10  1   8  30   80
    22211   23   15  1  12  60  200
    32111  113   20  1  16  90  320
    41111   14   14  1  10  45  140
   221111   24   21  1  15  90  350
   311111   15   22  1  12  63  224
  1111111    8   19  1   9  45  165
  2111111   16   26  1  14  84  336
  -------  ---   -- -- --- --- ----
              Sums: 22 185 810 2580 ...

Links

Crossrefs

Cf. A000041, A000712, A000716, A023003, A060850, A128627, A128629.

Programs

  • Maple
    with (numtheory): b:=proc(n) option remember; local d, j; `if` (n=0, 1, add (add (d, d=divisors(j)) *b(n-j), j=1..n)/n) end: A:= proc (n) option remember; local k; `if` (n=0, x, expand (add (b(k-1) *A(n-k) *x^(k-1), k=1..n))) end: a:= n-> coeftayl (A(n+8), x=0, 9): seq(a(n), n=1..40); # Alois P. Heinz, Oct 16 2008
# second Maple program:
a:= n-> n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 17 2008
  • Mathematica
    LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {22, 185, 810, 2580, 6765, 15525, 32305, 62337, 113265}, 30] (* Jean-François Alcover, Mar 07 2021 *)

Formula

From Alois P. Heinz, Oct 17 2008: (Start)
G.f.: x*(x-2)*(2*x^5-14*x^4+35*x^3-32*x^2-x+11)/(x-1)^9.
a(n) = n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320. (End)

Extensions

More terms from Alois P. Heinz, Oct 16 2008

A165187 a(n) = n^3*(n+1)^2*(n+2)/12.

Original entry on oeis.org

1, 24, 180, 800, 2625, 7056, 16464, 34560, 66825, 121000, 207636, 340704, 538265, 823200, 1224000, 1775616, 2520369, 3508920, 4801300, 6468000, 8591121, 11265584, 14600400, 18720000, 23765625, 29896776, 37292724, 46154080, 56704425, 69192000, 83891456, 101105664
Offset: 1

Views

Author

Alford Arnold, Sep 06 2009

Keywords

Comments

a(n) is row 30 of Table A128629 and can be generated by multiplying rows
two, three and five (or any other combination of rows with a row number product of 30).

Examples

			1,2,3,4,5, ... (A000027) times 1,3,6,10,15, ... (A000217) times 1,4,10,20,35, ... (A000292) yields 1,24,180,800, ...

Links

Crossrefs

Cf. A000027, A000217, A000292, A128629.

Programs

  • Mathematica
    a[n_] := n^3*(n+1)^2*(n+2)/12; Array[a, 35] (* Amiram Eldar, Feb 13 2023 *)

Formula

a(n) = A000027(n)*A000217(n)*A000292(n) = A128629(30,n).
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
G.f.: -x*(1+17*x+33*x^2+9*x^3)/(x-1)^7.
From Amiram Eldar, Feb 13 2023: (Start)
Sum_{n>=1} 1/a(n) = 153/4 - 9*Pi^2/2 + 6*zeta(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 48*log(2) - 141/4 - Pi^2/4 + 9*zeta(3)/2. (End)

Extensions

Edited and extended by R. J. Mathar, Sep 09 2009
Showing 1-3 of 3 results.

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