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

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

Author

Alford Arnold, Apr 03 2007

Keywords

• nonn
• uned

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

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