A164864 - OEIS

%I #40 Nov 05 2024 15:12:14

%S 1,1,2,5,15,52,203,877,4140,21147,115975,678569,4213530,27641927,

%T 190829797,1381367941,10448276360,82285618467,672294831619,

%U 5676711562593,49344452550230,439841775811967,4005444732928641,37136385907400125,349459367068932740

%N Number of ways of placing n labeled balls into 10 indistinguishable boxes; word structures of length n using a 10-ary alphabet.

%H Alois P. Heinz, <a href="/A164864/b164864.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 Dmytro S. Inosov and Emil Vlasák, <a href="https://arxiv.org/abs/2410.21427">Cryptarithmically unique terms in integer sequences</a>, arXiv:2410.21427 [math.NT], 2024. See pp. 3-4, 16-18.

%H N. Moreira and R. Reis, <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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Set_partition">Partition of a set</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (46,-906,9996,-67809,291774,-790964,1290824,-1136160,403200).

%F a(n) = Sum_{k=0..10} Stirling2 (n,k).

%F a(n) = ceiling(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800).

%F G.f.: (148329*x^9 -613453*x^8 +855652*x^7 -596229*x^6 +240065*x^5 -59410*x^4 +9177*x^3 -862*x^2 +45*x-1) / ((10*x-1) *(8*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).

%F a(n) <= A000110(n) with equality only for n <= 10.

%p # First program:

%p a:= n-> ceil(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800): seq(a(n), n=0..25);

%p # second program:

%p a:= n-> add(Stirling2(n, k), k=0..10): seq(a(n), n=0..25);

%t Table[Sum[StirlingS2[n,k],{k,0,10}],{n,0,30}] (* _Harvey P. Dale_, Nov 22 2023 *)

%Y Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164863.

%Y A row of the array in A278984.

%K easy,nonn

%O 0,3

%A _Alois P. Heinz_, Aug 28 2009