OFFSET
0,5
COMMENTS
a(n+1) = inverse binomial transform of A013999 = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*A013999(k). - Emanuele Munarini, Jul 01 2013
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..450
FORMULA
a(n) = Sum_{s=1..n} (-1)^(n-s)*s!*C(s+n-1,2s-1) if n>=1, where C(a,b)=binomial(a,b). - R. J. Mathar, Jul 13 2007
G.f.: Q(0) where Q(k) = 1 + (2*k + 1)*x/( (1+x)^2- 2*x*(1+x)^2*(k+1)/(2*x*(k+1) + (1+x)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 08 2013
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Oct 31 2017
MAPLE
A127548 := proc(n) if n = 0 then 1 ; else add(factorial(s)*(-1)^(n-s)*binomial(s+n-1, 2*s-1), s=1..n) ; fi ; end: for n from 0 to 20 do printf("%d, ", A127548(n)) ; od ; # R. J. Mathar, Jul 13 2007
MATHEMATICA
nn = 21; CoefficientList[Series[Sum[n!*(x/(1 + x)^2)^n, {n, 0, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Sep 04 2016 *)
PROG
(Python)
import math
def binomial(n, m):
a=1
for k in range(n-m+1, n+1):
a *= k
return a//math.factorial(m)
def A127548(n):
if n == 0:
return 1
a=0
for s in range(1, n+1):
a += (-1)**(n-s)*binomial(s+n-1, 2*s-1)*math.factorial(s)
return a
for n in range(30):
print(A127548(n))
# R. J. Mathar, Oct 20 2009
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jun 27 2007
EXTENSIONS
More terms from R. J. Mathar, Jul 13 2007
More terms from R. J. Mathar, Oct 20 2009
STATUS
approved