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A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds. 1

%I #23 Mar 03 2024 14:36:03

%S 22,185,810,2580,6765,15525,32305,62337,113265,195910,325193,521235,

%T 810654,1228080,1817910,2636326,3753600,5256711,7252300,9869990,

%U 13266099,17627775,23177583,30178575,38939875,49822812,63247635

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

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

%H Alois P. Heinz, <a href="/A129126/b129126.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

%F From _Alois P. Heinz_, Oct 17 2008: (Start)

%F G.f.: x*(x-2)*(2*x^5-14*x^4+35*x^3-32*x^2-x+11)/(x-1)^9.

%F a(n) = n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320. (End)

%e From A128629 we can construct the table below:

%e Deg # Associated sequence

%e ------- --- -------------------

%e 8 1 1 1 2 3 4

%e 44 2 3 1 3 6 10

%e 53 11 4 1 4 9 16

%e 62 11 4 1 4 9 16

%e 71 11 4 1 4 9 16

%e 332 12 6 1 6 18 40

%e 422 12 6 1 6 18 40

%e 431 111 8 1 8 27 64

%e 521 111 8 1 8 27 64

%e 611 12 6 1 6 18 40

%e 2222 4 7 1 5 15 35

%e 3221 112 12 1 12 54 160

%e 3311 22 9 1 9 36 100

%e 4211 112 12 1 12 54 160

%e 5111 13 10 1 8 30 80

%e 22211 23 15 1 12 60 200

%e 32111 113 20 1 16 90 320

%e 41111 14 14 1 10 45 140

%e 221111 24 21 1 15 90 350

%e 311111 15 22 1 12 63 224

%e 1111111 8 19 1 9 45 165

%e 2111111 16 26 1 14 84 336

%e ------- --- -- -- --- --- ----

%e Sums: 22 185 810 2580 ...

%p 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

%p # second Maple program:

%p a:= n-> n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320:

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 17 2008

%t 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 *)

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

%K nonn,uned

%O 1,1

%A _Alford Arnold_, Apr 03 2007

%E More terms from _Alois P. Heinz_, Oct 16 2008

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)