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A236921 Number of n-permutations which fix at least one odd prefix. 1
0, 0, 1, 2, 10, 32, 232, 992, 10096, 53408, 727360, 4569536, 79501696, 578101376, 12337163008, 101945840384, 2582987522560, 23913303638528, 701604503968768, 7194776722623488, 239847438803052544, 2698941227297687552, 100744097104231198720, 1234263151585971974144, 50993324690816940089344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

Warren D. Smith, Postings to Math Fun Mailing List, Feb 06 2014 - Feb 08 2014.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

FORMULA

a(0)=a(1)=0; a(n) = Sum_(k=1,3,5,..., whichever is odd among {n-1, n-2}) (k!-a(k))*(n-k)!.

To see why this recurrence holds, enumerate all the a(n) permutations of {1,2,3,...,n} which fix an odd prefix. They are:

perms of form   their count

1...            (n-1)!

(123)...        (3!-2)*(n-3)!  where we count only the ones not of the preceding form;  that is, (3!-a(3))*(n-3)!

(12345)...      (5!-a(5))*(n-5)! where again count only those not of preceding two forms,

and so on. [Warren D. Smith]

a(n) ~ (3+(-1)^n)/2 * (n-1)!. - Vaclav Kotesovec, Feb 15 2014

MAPLE

F := array(1..66); F[1] := 0;

F[2] := 1;

for n from  3  to        66  do

F[n] := sum( ((2*j+1)! - F[2*j+1]) * (n-(2*j+1))!, j=0 .. (n-2)/2 );

od; # From Warren D. Smith, Feb 12 2014

CROSSREFS

Sequence in context: A151019 A004028 A263839 * A316644 A080668 A062453

Adjacent sequences:  A236918 A236919 A236920 * A236922 A236923 A236924

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 11 2014

STATUS

approved

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Last modified November 14 01:24 EST 2019. Contains 329108 sequences. (Running on oeis4.)