

A122649


Difference between the double factorial of the nth nonnegative odd number and the double factorial of the nth nonnegative even number.


6



0, 1, 7, 57, 561, 6555, 89055, 1381905, 24137505, 468934515, 10033419375, 234484536825, 5943863027025, 162446292283275, 4761954230608575, 149048910271886625, 4961463912662882625, 175022432901300859875
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OFFSET

1,3


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..410


FORMULA

a(n) = (2*n  1)!!  (2*n  2)!! = A006882(2*n  1)  A000165(n  1).
From Peter Bala, Jun 22 2016: (Start)
a(1) = 0, a(2) = 1 and for n >= 3, a(n) = (4*n  5)*a(n1)  (2*n  4)*(2*n  3)*a(n2).
E.g.f. assuming an offset of 0: A(x) = 1/(1  2*x)^(3/2)  1/(1  2*x) = x + 7*x^2/2! + 57*x^3/3! + ....
A( Sum_{n >= 1} n^(n2)*x^n/n! ) = Sum_{n >= 1} n^(n+1)*x^n/n!.
Series reversion (A(x)) = 1/2*Sum_{n >= 1} (1)^(n+1)*1/(n+1)*
binomial(3*n + 1,n)*x^n. Cf. A006013.(End)
E.g.f.: 1 + 1/sqrt(12*x) + log(12*x)/2.  Ilya Gutkovskiy, Jun 23 2016


EXAMPLE

a(1) = 0, since 1!!  0!! = 1  1 = 0, where the usual convention 0!! = 1 has been heeded. Note that 1 is the first nonnegative odd and 0 the first nonnegative even number.
a(4) = 57, since 7!!  6!! = 1*3*5*7  6*4*2*1 = 105  48 = 57.


MAPLE

for n from 1 to 24 do: l[n]:=product(2*k1, k=1..n); od: r[1]:=1; for n from 2 to 24 do: r[n]:=product(2*k, k=1..n1); od; for k from 1 to 24 do: a[k]:=l[k]r[k]; od;


MATHEMATICA

#[[2]]#[[1]]&/@Partition[Range[0, 40]!!, 2] (* Harvey P. Dale, Feb 19 2013 *)
Rest[Range[0, 100]! CoefficientList[Series[1 + 1/Sqrt[1  2 x] + Log[1  2 x]/2, {x, 0, 800}], x]] (* Vincenzo Librandi, Jun 24 2016 *)


CROSSREFS

Cf. A006882, A000165, A006013.
Sequence in context: A248168 A176733 A062192 * A051846 A231540 A316442
Adjacent sequences: A122646 A122647 A122648 * A122650 A122651 A122652


KEYWORD

easy,nonn


AUTHOR

Peter C. Heinig (algorithms(AT)gmx.de), Sep 21 2006


STATUS

approved



