A122649 Difference between the double factorial of the n-th nonnegative odd number and the double factorial of the n-th nonnegative even number.

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

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(n-1) - (2*n - 4)*(2*n - 3)*a(n-2).

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^(n-2)*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(1-2*x) + log(1-2*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*k-1, k=1..n); od: r[1]:=1; for n from 2 to 24 do: r[n]:=product(2*k, k=1..n-1); 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 A349116

Adjacent sequences: A122646 A122647 A122648 * A122650 A122651 A122652

Keyword

easy,nonn

Author

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

Status

approved