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A129893 a(n) = s!/(s-n)! where s = (n*(n+1)/2)+1.

%I A129893 #33 Jan 05 2025 19:51:38
%S A129893 1,2,12,210,7920,524160,53721360,7866331200,1556675366400,
%T A129893 399790821830400,129210868410624000,51295616536721356800,
%U A129893 24529502681864788608000,13903600298770901182464000
%N A129893 a(n) = s!/(s-n)! where s = (n*(n+1)/2)+1.
%C A129893 Bread Shop Open!. We have a loaf of bread which has a kernel of corns irregularly inside. We cut the loaf n times getting the maximal number (s, see A000124) of pieces and distribute one piece to each of n people. The remaining pieces of bread will be the prize for the winner. The sequence gives the number of cases when n pieces are distributed to n persons.
%D A129893 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A129893 H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
%D A129893 N. Reading, On the structure of Bruhat Order, Ph.D. dissertation, University of Minnesota, anticipated 2002.
%D A129893 A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
%D A129893 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%D A129893 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
%D A129893 A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964)
%H A129893 Reinhard Zumkeller, <a href="/A129893/b129893.txt">Table of n, a(n) for n = 0..120</a>
%H A129893 A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, arXiv:math/0112281 [math.CO], 2001; Annals. Combin., 7 (2003), 1-14; see Example 3.5.
%H A129893 L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n25">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
%H A129893 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H A129893 D. A. Lind, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-4/lind.pdf">On a class of nonlinear binomial sums</a>, Fib. Quart., 3 (1965), 292-298.
%H A129893 D. J. Price, <a href="http://www.jstor.org/stable/3609091">Some unusual series occurring in n-dimensional geometry</a>, Math. Gaz., 30 (1946), 149-150.
%H A129893 R. Simion and F. W. Schmidt, <a href="https://doi.org/10.1016/S0195-6698(85)80052-4">Restricted permutations</a>, European J. Combin., 6, 383-406, 1985.
%F A129893 a(n) = sPn, where s=(n*(n+1)/2)+1.
%e A129893 a(2)=12 s=4,n=2 because we can write 12=4*3.
%e A129893 a(3)=210 s=7,n=3 because we can write 210=7*6*5.
%t A129893 Table[s=(n(n+1))/2+1;s!/(s-n)!,{n,0,20}] (* _Harvey P. Dale_, Nov 15 2012 *)
%t A129893 #[[1]]!/(#[[1]]-#[[2]])!&/@With[{nn=20},Thread[{Accumulate[ Range[0,nn]]+ 1,Range[0,nn]}]] (* _Harvey P. Dale_, Sep 12 2015 *)
%o A129893 (Haskell)
%o A129893 a129893 n = a129893_list !! n
%o A129893 a129893_list = 1 : zipWith div (tail fs) fs where
%o A129893    fs = map a000142 a000124_list
%o A129893 -- _Reinhard Zumkeller_, Oct 03 2012
%Y A129893 Cf. A000124, A107868, A129933.
%Y A129893 Cf. A000142.
%K A129893 nonn,nice
%O A129893 0,2
%A A129893 Kim Dong Seok (Go Jae Song, Nam Dae Young) from KNU (gjs0419(AT)nate.com), Jun 04 2007
%E A129893 Typo fixed in a(13) by _Reinhard Zumkeller_, Oct 03 2012