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A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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)

EXAMPLE

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 ...

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-Fran├žois Alcover, Mar 07 2021 *)

CROSSREFS

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

Sequence in context: A248495 A248496 A229367 * A231749 A072040 A022682

Adjacent sequences:  A129123 A129124 A129125 * A129127 A129128 A129129

KEYWORD

nonn,uned

AUTHOR

Alford Arnold, Apr 03 2007

EXTENSIONS

More terms from Alois P. Heinz, Oct 16 2008

STATUS

approved

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Last modified September 26 09:15 EDT 2021. Contains 347664 sequences. (Running on oeis4.)