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.
%I A129126 #23 Mar 03 2024 14:36:03
%S A129126 22,185,810,2580,6765,15525,32305,62337,113265,195910,325193,521235,
%T A129126 810654,1228080,1817910,2636326,3753600,5256711,7252300,9869990,
%U A129126 13266099,17627775,23177583,30178575,38939875,49822812,63247635
%N A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds.
%C A129126 A slightly different method of calculating this sequence is described in A128627.
%H A129126 Alois P. Heinz, <a href="/A129126/b129126.txt">Table of n, a(n) for n = 1..1000</a>
%H A129126 <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 A129126 From _Alois P. Heinz_, Oct 17 2008: (Start)
%F A129126 G.f.: x*(x-2)*(2*x^5-14*x^4+35*x^3-32*x^2-x+11)/(x-1)^9.
%F A129126 a(n) = n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320. (End)
%e A129126 From A128629 we can construct the table below:
%e A129126 Deg # Associated sequence
%e A129126 ------- --- -------------------
%e A129126 8 1 1 1 2 3 4
%e A129126 44 2 3 1 3 6 10
%e A129126 53 11 4 1 4 9 16
%e A129126 62 11 4 1 4 9 16
%e A129126 71 11 4 1 4 9 16
%e A129126 332 12 6 1 6 18 40
%e A129126 422 12 6 1 6 18 40
%e A129126 431 111 8 1 8 27 64
%e A129126 521 111 8 1 8 27 64
%e A129126 611 12 6 1 6 18 40
%e A129126 2222 4 7 1 5 15 35
%e A129126 3221 112 12 1 12 54 160
%e A129126 3311 22 9 1 9 36 100
%e A129126 4211 112 12 1 12 54 160
%e A129126 5111 13 10 1 8 30 80
%e A129126 22211 23 15 1 12 60 200
%e A129126 32111 113 20 1 16 90 320
%e A129126 41111 14 14 1 10 45 140
%e A129126 221111 24 21 1 15 90 350
%e A129126 311111 15 22 1 12 63 224
%e A129126 1111111 8 19 1 9 45 165
%e A129126 2111111 16 26 1 14 84 336
%e A129126 ------- --- -- -- --- --- ----
%e A129126 Sums: 22 185 810 2580 ...
%p A129126 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 A129126 # second Maple program:
%p A129126 a:= n-> n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320:
%p A129126 seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 17 2008
%t A129126 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 A129126 Cf. A000041, A000712, A000716, A023003, A060850, A128627, A128629.
%K A129126 nonn,uned
%O A129126 1,1
%A A129126 _Alford Arnold_, Apr 03 2007
%E A129126 More terms from _Alois P. Heinz_, Oct 16 2008