%I #6 Oct 14 2017 10:51:28

%S 7087261,492006708,19423259316,574637640288,14193955791576,

%T 309660911167464,6171397007611848,114853532449557600,

%U 2026594842428425320,34277110454602760762,560261324259420037164,8904738970375872782112,138290600270036591006520,2106511986693346884064584

%N Number of compositions of n where each part i is marked with a word of length i over a nonary alphabet whose letters appear in alphabetical order and all nine letters occur at least once in the composition.

%H Alois P. Heinz, <a href="/A293586/b293586.txt">Table of n, a(n) for n = 9..885</a>

%F a(n) = 90*a(n-1) - 3720*a(n-2) + 94140*a(n-3) - 1642368*a(n-4) + 21111972*a(n-5) - 208936444*a(n-6) + 1643906838*a(n-7) - 10544413816*a(n-8) + 56273496182*a(n-9) - 254124223400*a(n - 10) + 984813733064*a(n - 11) - 3313818868728*a(n - 12) + 9777617820932*a(n - 13) - 25505157099056*a(n - 14) + 59222241227144*a(n - 15) - 123105458091224*a(n - 16) + 230174411303404*a(n - 17) - 388610578141384*a(n - 18) + 594331344450528*a(n - 19) - 825476563250976*a(n - 20) + 1043293124084592*a(n - 21) - 1201650502768408*a(n - 22) + 1262594519234968*a(n - 23) - 1210928179506120*a(n - 24) + 1060266691901408*a(n - 25) - 847323181595664*a(n - 26) + 617639581793392*a(n - 27) - 410205458302944*a(n - 28) + 247843724510640*a(n - 29) - 135949707500048*a(n - 30) + 67526545242016*a(n - 31) - 30273460576096*a(n - 32) + 12201462236512*a(n - 33) - 4399521714368*a(n - 34) + 1410734015840*a(n - 35) - 399326676032*a(n - 36) + 98870585152*a(n - 37) - 21165129088*a(n - 38) + 3859085248*a(n - 39) - 587513600*a(n - 40) + 72658304*a(n - 41) - 7011968*a(n - 42) + 495360*a(n - 43) - 22784*a(n - 44) + 512*a(n - 45). - _Vaclav Kotesovec_, Oct 14 2017

%p b:= proc(n, k) option remember; `if`(n=0, 1,

%p add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))

%p end:

%p a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(9):

%p seq(a(n), n=9..30);

%Y Column k=9 of A261781.

%K nonn

%O 9,1

%A _Alois P. Heinz_, Oct 12 2017