A203641 Number of arrays of n 0..10 integers with new values introduced in order 0..10 but otherwise unconstrained.

1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213596, 27644358, 190895863, 1382847419, 10477213268, 82797679445, 680685836527, 5806124780384, 51245294979716, 466668627500968, 4371727233798927, 42000637216351225

Offset: 1

Author

R. H. Hardin, Jan 04 2012

Keywords

• nonn
• easy

Comments

From Danny Rorabaugh, Mar 03 2015: (Start)

a(n) is also the number of ways of placing n labeled balls into 11 indistinguishable boxes.

a(n) is also the number of word structures of length n using an 11-ary alphabet.

(End)

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Eric Weisstein's World of Mathematics, Set Partition.
• Index entries for linear recurrences with constant coefficients, signature (56, -1365, 19020, -167223, 965328, -3686255, 9133180, -13926276, 11655216, -3991680).

Crossrefs

Column k=10 of A203647.

Cf. A056272, A056273, A164863, A164864.

Programs

Maple

f:= n -> add(Stirling2(n,k),k=1..11):
map(f, [$1..100]); # Robert Israel, Aug 08 2016

PARI

a(n) = sum(k=1,11,stirling(n,k, 2)); \\ Michel Marcus, Mar 03 2015

Formula

Empirical: a(n) = 56*a(n-1) -1365*a(n-2) +19020*a(n-3) -167223*a(n-4) +965328*a(n-5) -3686255*a(n-6) +9133180*a(n-7) -13926276*a(n-8) [phone]*a(n-9) -3991680*a(n-10).

a(n) = Sum_{k=1..11} stirling2(n,k). - Danny Rorabaugh, Mar 03 2015

G.f.: Sum_{k=1..11} Product_{j=1..k} x/(1-j*x). This confirms the empirical recurrence. - Robert Israel, Aug 08 2016