A164863 Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678514, 4211825, 27602602, 190077045, 1368705291, 10254521370, 79527284317, 635182667816, 5199414528808, 43426867585575, 368654643520692, 3170300933550687, 27542984610086665, 241205285284001240
Offset: 0
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"
- Moreira, N.; Reis, R. "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
- Index entries for linear recurrences with constant coefficients, signature (37, -574, 4858, -24409, 74053, -131256, 122652, -45360).
Crossrefs
Programs
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Maple
# first program: 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); # second program: a:= n-> add(Stirling2(n, k), k=0..9): seq(a(n), n=0..25);
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Mathematica
Table[Sum[StirlingS2[n, k], {k, 0, 9}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
Formula
a(n) = Sum_{k=0..9} stirling2 (n,k).
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).
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)).
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