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A203641 Number of arrays of n 0..10 integers with new values introduced in order 0..10 but otherwise unconstrained.

%I A203641 #35 Feb 16 2025 08:33:16
%S A203641 1,2,5,15,52,203,877,4140,21147,115975,678570,4213596,27644358,
%T A203641 190895863,1382847419,10477213268,82797679445,680685836527,
%U A203641 5806124780384,51245294979716,466668627500968,4371727233798927,42000637216351225
%N A203641 Number of arrays of n 0..10 integers with new values introduced in order 0..10 but otherwise unconstrained.
%C A203641 From _Danny Rorabaugh_, Mar 03 2015: (Start)
%C A203641 a(n) is also the number of ways of placing n labeled balls into 11 indistinguishable boxes.
%C A203641 a(n) is also the number of word structures of length n using an 11-ary alphabet.
%C A203641 (End)
%H A203641 R. H. Hardin, <a href="/A203641/b203641.txt">Table of n, a(n) for n = 1..210</a>
%H A203641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SetPartition.html">Set Partition</a>.
%H A203641 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (56, -1365, 19020, -167223, 965328, -3686255, 9133180, -13926276, 11655216, -3991680).
%F A203641 Empirical: a(n) = 56*a(n-1) -1365*a(n-2) +19020*a(n-3) -167223*a(n-4) +965328*a(n-5) -3686255*a(n-6) +9133180*a(n-7) -13926276*a(n-8) +11655216*a(n-9) -3991680*a(n-10).
%F A203641 a(n) = Sum_{k=1..11} stirling2(n,k). - _Danny Rorabaugh_, Mar 03 2015
%F A203641 G.f.: Sum_{k=1..11} Product_{j=1..k} x/(1-j*x).  This confirms the empirical recurrence. - _Robert Israel_, Aug 08 2016
%p A203641 f:= n -> add(Stirling2(n,k),k=1..11):
%p A203641 map(f, [$1..100]); # _Robert Israel_, Aug 08 2016
%o A203641 (PARI) a(n) = sum(k=1,11,stirling(n,k, 2)); \\ _Michel Marcus_, Mar 03 2015
%Y A203641 Column k=10 of A203647.
%Y A203641 Cf. A056272, A056273, A164863, A164864.
%K A203641 nonn,easy
%O A203641 1,2
%A A203641 _R. H. Hardin_, Jan 04 2012