This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001564 M2972 N1202 #78 Aug 24 2025 02:05:56
%S A001564 1,3,14,78,504,3720,30960,287280,2943360,33022080,402796800,
%T A001564 5308934400,75203251200,1139544806400,18394619443200,315149522688000,
%U A001564 5711921639424000,109196040425472000,2196014181064704000,46346783255764992000,1024251745442365440000
%N A001564 2nd differences of factorial numbers.
%C A001564 a(n) is also the number of isolated fixed points (i.e. adjacent fixed points are not isolated) in all permutations of [n+2]. Example: a(2)=14 because we have (the isolated fixed points are marked) 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 42'13, 2314', 243'1, 32'14', 32'41. - _Emeric Deutsch_, Apr 18 2009
%C A001564 The average of the first n terms is n factorial. - _Franklin T. Adams-Watters_, May 20 2010
%C A001564 Number of blocks in all permutations of [n+1]. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67. Example: a(2)=14 because the permutations of [3], separated into blocks, are 123, 1-3-2, 2-1-3, 23-1, 3-12, 3-2-1 with 1+3+3+2+2+3=14 blocks. - _Emeric Deutsch_, Jul 12 2010
%C A001564 a(n) equals n+1 times the permanent of the (n+1) X (n+1) matrix with 1/(n+1) in the top right corner and 1's everywhere else. - _John M. Campbell_, May 25 2011
%C A001564 Number of permutations s of [n+2] where 2 designated elements are not mapped to themselves, e.g., s(1) != 1 and s(2) != 2. See Janjić article. - _Benjamin Schreyer_, May 07 2025
%D A001564 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001564 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001564 T. D. Noe, <a href="/A001564/b001564.txt">Table of n, a(n) for n = 0..100</a>
%H A001564 A. van Heemert, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002177749">Cyclic permutations with sequences and related problems</a>, J. Reine Angew. Math., 198 (1957), 56-72.
%H A001564 Milan Janjić, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulae for Some Functions on Finite Sets</a>
%H A001564 A. N. Myers, <a href="http://dx.doi.org/10.1006/jcta.2002.3279">Counting permutations by their rigid patterns</a>, J. Combin. Theory, A 99 (2002), 345-357.
%H A001564 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A001564 a(n) = (n^2 + n + 1)*n! = A002061(n-1)*A000142(n). - _Mitch Harris_, Jul 10 2008
%F A001564 E.g.f.: (1+x^2)/(1-x)^3.
%F A001564 a(n) = A001563(n+1) - A001563(n). - _Robert Israel_, Apr 13 2015
%F A001564 a(n) = A306209(n+2,n). - _Alois P. Heinz_, Jan 29 2019
%F A001564 D-finite with recurrence a(n) +(-n-3)*a(n-1) +(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 01 2022
%p A001564 seq(factorial(n)*(n^2+n+1), n = 0 .. 20); # _Emeric Deutsch_, Apr 18 2009
%t A001564 Range[0,20]! CoefficientList[Series[(1+x^2)/(1-x)^3,{x,0,20}],x]
%t A001564 Differences[Range[0, 25]!, 2] (* _Paolo Xausa_, Jul 17 2025 *)
%o A001564 (PARI) Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ _Michel Marcus_, Apr 10 2015
%o A001564 (Magma) [(n^2+n+1)*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Apr 10 2015
%Y A001564 Cf. A000142, A001563, A002061, A010027, A047920, A306209.
%K A001564 nonn,easy,changed
%O A001564 0,2
%A A001564 _N. J. A. Sloane_
%E A001564 Comment edited by _Franklin T. Adams-Watters_, May 20 2010