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A122649 Difference between the double factorial of the n-th nonnegative odd number and the double factorial of the n-th nonnegative even number. 6
0, 1, 7, 57, 561, 6555, 89055, 1381905, 24137505, 468934515, 10033419375, 234484536825, 5943863027025, 162446292283275, 4761954230608575, 149048910271886625, 4961463912662882625, 175022432901300859875 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
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
Sequence in context: A248168 A176733 A062192 * A051846 A231540 A349116
KEYWORD
easy,nonn
AUTHOR
Peter C. Heinig (algorithms(AT)gmx.de), Sep 21 2006
STATUS
approved

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Last modified February 28 17:13 EST 2024. Contains 370400 sequences. (Running on oeis4.)