A164864 Number of ways of placing n labeled balls into 10 indistinguishable boxes; word structures of length n using a 10-ary alphabet.

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213530, 27641927, 190829797, 1381367941, 10448276360, 82285618467, 672294831619, 5676711562593, 49344452550230, 439841775811967, 4005444732928641, 37136385907400125, 349459367068932740

Offset: 0

Author

Alois P. Heinz, Aug 28 2009

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3-4, 16-18.
• N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
• Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
• Eric Weisstein's World of Mathematics, Set Partition
• Wikipedia, Partition of a set
• Index entries for linear recurrences with constant coefficients, signature (46,-906,9996,-67809,291774,-790964,1290824,-1136160,403200).

Crossrefs

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

A row of the array in A278984.

Programs

Maple

# First program:
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);
# second program:
a:= n-> add(Stirling2(n, k), k=0..10): seq(a(n), n=0..25);

Mathematica

Table[Sum[StirlingS2[n,k],{k,0,10}],{n,0,30}] (* Harvey P. Dale, Nov 22 2023 *)

Formula

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

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).

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)).

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