%I #33 Jul 04 2024 09:29:07
%S 1,1,2,5,15,52,203,877,4140,21147,115974,678514,4211825,27602602,
%T 190077045,1368705291,10254521370,79527284317,635182667816,
%U 5199414528808,43426867585575,368654643520692,3170300933550687,27542984610086665,241205285284001240
%N Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.
%H Alois P. Heinz, <a href="/A164863/b164863.txt">Table of n, a(n) for n = 0..1000</a>
%H Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H Moreira, N.; Reis, R. "<a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>", Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H Pierpaolo Natalini, Paolo Emilio Ricci, <a href="https://doi.org/10.3390/axioms7040071">New Bell-Sheffer Polynomial Sets</a>, Axioms 2018, 7(4), 71.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SetPartition.html">Set Partition</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (37, -574, 4858, -24409, 74053, -131256, 122652, -45360).
%F a(n) = Sum_{k=0..9} stirling2 (n,k).
%F a(n) = ceiling (103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880).
%F G.f.: (16687*x^8 -67113*x^7 +88620*x^6 -56993*x^5 +20529*x^4 -4353*x^3 +539*x^2 -36*x+1) / ((9*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).
%F G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=9. - _Robert A. Russell_, Apr 25 2018
%p # first program:
%p a:= n-> ceil(103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880): seq(a(n), n=0..25);
%p # second program:
%p a:= n-> add(Stirling2(n, k), k=0..9): seq(a(n), n=0..25);
%t Table[Sum[StirlingS2[n, k], {k, 0, 9}], {n, 0, 30}] (* _Robert A. Russell_, Apr 25 2018 *)
%Y Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164864.
%Y A row of the array in A278984.
%K easy,nonn
%O 0,3
%A _Alois P. Heinz_, Aug 28 2009