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A127548
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O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.
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3
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1, 1, 0, 1, 4, 19, 112, 771, 6088, 54213, 537392, 5867925, 69975308, 904788263, 12607819040, 188341689287, 3002539594128, 50878366664393, 913161208490016, 17304836525709097, 345279674107957524, 7235298537356113339
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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nn = 21; CoefficientList[Series[Sum[n!*(x/(1 + x)^2)^n, {n, 0, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Sep 04 2016 *)
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PROG
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(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)
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):
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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