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

0,3

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"

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).

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)).

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 *)

Crossrefs

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

A row of the array in A278984.

Sequence in context: A287281 A287259 A287671 * A366776 A192866 A229227

Adjacent sequences: A164861 A164862 A164863 * A164865 A164866 A164867

Keyword

easy,nonn

Author

Alois P. Heinz, Aug 28 2009

Status

approved