A271744 Number of set partitions of [n] such that 5 is the largest element of the last block.

30, 59, 139, 365, 1039, 3149, 10039, 33365, 114799, 406589, 1475239, 5460965, 20550559, 78375629, 302129239, 1174610165, 4597146319, 18085554269, 71434828039, 283020552965, 1123932750079, 4471296246509, 17811907987639, 71027855481365, 283452201569839

Offset: 5

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000
• Wikipedia, Partition of a set
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Column k=5 of A271466.

Programs

Magma

I:=[30,59,139,365,1039]; [n le 5 select I[n] else 10*Self(n-1)-35*Self(n-2)+50*Self(n-3)-24*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 13 2016

Mathematica

Join[{30}, LinearRecurrence[{10, -35, 50, -24}, {59, 139, 365, 1039}, 30]] (* Vincenzo Librandi, Apr 13 2016 *)

PARI

Vec(x^5*(30 - 241*x + 599*x^2 - 460*x^3 + 24*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Dec 24 2017

Formula

G.f.: x^5*(24*x^4-460*x^3+599*x^2-241*x+30)/Product_{j=1..4} (j*x-1).

From Colin Barker, Dec 24 2017: (Start)

a(n) = 8 + 13*2^(n-5) + 7*3^(n-5) + 4^(n-5) for n>5.

a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n>9.

(End)