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A224271
Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.
3
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
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]
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 02 2013
STATUS
approved