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A010551 Multiply successively by 1,1,2,2,3,3,4,4,..., n >= 1, a(0) = 1. 19
1, 1, 1, 2, 4, 12, 36, 144, 576, 2880, 14400, 86400, 518400, 3628800, 25401600, 203212800, 1625702400, 14631321600, 131681894400, 1316818944000, 13168189440000, 144850083840000, 1593350922240000, 19120211066880000, 229442532802560000, 2982752926433280000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From Emeric Deutsch, Dec 14 2008: (Start)

Number of permutations of {1,2,...,n-1} having a single run of odd entries. Example: a(5)=12 because we have 1324,1342,3124,3142,2134,4132,2314,4312, 2413, 4213, 2431 and 4231.

a(n)=A152666(n-1,1). (End)

a(n+1) gives the permanent of the n X n matrix whose (i,j)-element is i+j-1 modulo 2. - John W. Layman, Jan 03 2011

From Daniel Forgues, May 20 2011: (Start)

a(0) = 1 since it is the empty product.

A010551(n-2), n >= 2, equal to (ceiling((n-2)/2))! * (floor((n-2)/2))!, gives the number of arrangements of n-2 entries from 2 to n-1, starting with an even entry and where the parity of adjacent entries alternates. This is the number of arrangements to investigate for row n of a prime pyramid (A051237). (End)

Partial products of A008619. - Reinhard Zumkeller, Apr 02 2012

Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb where a<b<c, cf. A210667 (equivalently under such transformations of the form abc <--> bac where a<b<c.) - Tom Roby, 15 May 2012

Row sums of A246117. - Peter Bala, Aug 15 2014

Sum_{n >= 1} 1/a(n) equals the constant A130820. - Peter Bala, Jul 02 2016

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Steven Linton, James Propp, Tom Roby, Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.

FORMULA

a(n) = [n/2]! * [(n+1)/2 ]! is the number of permutations p of {1, 2, 3, ..., n} such that for every i, i and p(i) have the same parity, i.e., p(i) - i is even. - Avi Peretz (njk(AT)netvision.net.il), Feb 22 2001

a(n) = n!/binomial(n, floor(n/2)). - Paul Barry, Sep 12 2004

G.f.: Sum_{n>=0} x^n/a(n) = besseli(0, 2*x) + x*besseli(1, 2*x). - Paul D. Hanna, Apr 07 2005

E.g.f.: 1/(1-x/2) + (1/2)/(1-x/2)*acos(1-x^2/2)/sqrt(1-x^2/4). - Paul D. Hanna, Aug 28 2005

G.f.: G(0)  where G(k) = 1 + (k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1) )) ; (continued fraction,3-step). - Sergei N. Gladkovskii, Nov 28 2012

Conjecture: 4*a(n) -2*a(n-1) -n*(n-1)*a(n-2)=0. - R. J. Mathar, Dec 03 2012

a(n) = a(n-1) * (a(n-2) + a(n-3)) / a(n-3) for all n>=3. - Michael Somos, Dec 29 2012

G.f.: 1 + x + x^2*(1 + x*(G(0) - 1)/(x-1)) where G(k) = 1 - (k+2)/(1-x/(x - 1/(1 - (k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-x/(x - 1/(1 - (k+1)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013

G.f.: 1 + x*G(0), where G(k)= 1 + x*(k+1)/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013

G.f.: Q(0), where Q(k)= 1 + x*(k+1)/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 144*x^7 + 576*x^8 + ...

For n = 7, a(n) = 1*1*2*2*3*3*4 (7 factors), which is 144. - Michael B. Porter, Jul 03 2016

MAPLE

A010551 := proc(n) option remember; if n <= 1 then 1 else A010551(n-1) *trunc( (n+1)/2 ); fi; end;

MATHEMATICA

FoldList[ Times, 1, Flatten@ Array[ {#, #} &, 11]] (* Robert G. Wilson v, Jul 14 2010 *)

PROG

(PARI) {a(n)=local(X=x+x*O(x^n)); 1/polcoeff(besseli(0, 2*X)+X*besseli(1, 2*X), n, x)} \\ Paul D. Hanna

(Haskell)

a010551 n = a010551_list !! n

a010551_list = scanl (*) 1 a008619_list

-- Reinhard Zumkeller, Apr 02 2012

(PARI) {a(n) = if( n<0, 0, (n\2)! * ((n+1)\2)!)}; /* Michael Somos, Dec 29 2012 */

CROSSREFS

Cf. A008619, A064044, A246117, A130820.

Column k=2 of A275062.

Sequence in context: A046993 A282165 A111942 * A276230 A003701 A255432

Adjacent sequences:  A010548 A010549 A010550 * A010552 A010553 A010554

KEYWORD

nonn

AUTHOR

Mark R. Diamond

STATUS

approved

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Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.