A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

1, 1, 3, 8, 28, 107, 459, 2151, 10931, 59700, 348146, 2155925, 14112377, 97266301, 703484851, 5323515156, 42040470092, 345670438963, 2953171501547, 26166317121747, 240047041176843, 2276607815242880, 22290187889601330, 225018607554567149, 2339331996135377345

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..576

Formula

E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.

a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - Alois P. Heinz, Apr 02 2013

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

From Alois P. Heinz, May 17 2023: (Start)

a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).

a(n) mod 2 = A131719(n+1). (End)

Example

a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 08 2015
# second Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
      b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n-1, 1$2):
seq(a(n), n=1..25);  # Alois P. Heinz, May 17 2023

Mathematica

nn=25; Drop[Range[0, nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]], x], x], {x, 0, nn}], x], 1]

Crossrefs

Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.

Sequence in context: A245892 A263103 A093356 * A135583 A317077 A009437

Adjacent sequences: A224268 A224269 A224270 * A224272 A224273 A224274

Keyword

nonn

Author

Geoffrey Critzer, Apr 02 2013

Status

approved