OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n} (s(n,k)*(-1)^n-k)*b(k)/2^(ceiling(n/2)-k) where the s(n,k) are the (signless) Stirling numbers of the first kind, and the b(n) are the Bell numbers.
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-1)^k*2^(floor(n/2)-2*k)*((2*k)!/k!)*i(n-2*k), where the i(n) are the involution numbers A000085.
Recurrence: a(n+2) = 2^even(n)*a(n+1) + (n+1)*a(n), where even(n) = 1 when n is even and even(n) = 0 otherwise.
G.f.: Sum_{n>=0} a(n)*x^n/(2^floor(n/2)*n!) = exp(x+x^2/4).
a(n) ~ 2^((-3 + (-1)^n)/4) * n^(n/2) / exp(n/2 + 1/2 - sqrt(2*n)) * (1+sqrt(2/n)/3). - Vaclav Kotesovec, May 24 2014
a(n) = I^(-n)*2^floor(n/2)*U(-n/2,1/2,-1), where U(a,b,x) is Kummer's hypergeometric function of the second kind. - Peter Luschny, May 24 2014
a(n) = numerator(a(n-1) + (a(n-2)/2)*(n-2)) for a(1)=0 and a(2)=1. - Terry D. Grant, May 04 2017
MAPLE
a := n -> I^(-n)*2^floor(n/2)*KummerU(-n/2, 1/2, -1);
seq(round(evalf(a(n), 32)), n = 0..26); # Peter Luschny, May 24 2014
MATHEMATICA
Table[Sum[StirlingS1[n, k] BellB[k]/2^(Ceiling[n/2] - k), {k, 0, n}], {n, 0, 100}]
PROG
(Maxima) makelist(sum(stirling1(n, k)*belln(k)/2^(ceiling(n/2)-k), k, 0, n), n, 0, 41);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, May 23 2014
STATUS
approved