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 A010551 Multiply successively by 1,1,2,2,3,3,4,4,..., n >= 1, a(0) = 1. 25
 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 E. K. Gnang, I. Wass, Growing graceful trees, arXiv:1808.05551 [math.CO], 2018. See proposition 1. 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) = floor(n/2)!*floor((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)*arccos(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 a(n) ~  sqrt(Pi*n) * n! / 2^(n + 1/2). - Vaclav Kotesovec, Oct 02 2018 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         procname(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) A010551(n)=(n\2)!*((n+1)\2)! \\ Michael Somos, Dec 29 2012, edited by M. F. Hasler, Nov 26 2017 (MAGMA) [Factorial(n div 2)*Factorial((n+1) div 2): n in [0..25]]; // Vincenzo Librandi Jan 17 2018 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,easy AUTHOR STATUS approved

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Last modified May 21 10:48 EDT 2019. Contains 323443 sequences. (Running on oeis4.)