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
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).
Programs
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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
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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
Comments