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A275518 Number of simplices in corner-cut triangulation of the n-cube.

%I A275518 #18 Sep 06 2023 15:25:23
%S A275518 1,2,5,16,67,364,2445,19296,173015,1728604,19011049,228124384,
%T A275518 2965598547,41518338684,622774990133,9964399645504,169394793547567,
%U A275518 3049106282938684,57933019373868897,1158660387473183616,24331868136927943019,535301099012395872028
%N A275518 Number of simplices in corner-cut triangulation of the n-cube.
%C A275518 This corrects the value of a(10) in A239911 published by Sallee in Discr. Math. 40. The correct value is for example given by Lee.
%H A275518 Andrew Howroyd, <a href="/A275518/b275518.txt">Table of n, a(n) for n = 1..200</a>
%H A275518 Carl W. Lee, <a href="http://dx.doi.org/10.1111/j.1749-6632.1985.tb14555.x">Triangulating the d-cube</a>, Annals of the New York Academy of Sciences 440 (1985): 205-211.
%H A275518 John F. Sallee, <a href="http://dx.doi.org/10.1016/0166-218X(82)90041-5">A note on minimal triangulations of an n-cube</a>,  Discrete Appl. Math. 4 (1982), no. 3, 211-215. MR0675850 (84g:52019)
%H A275518 John F. Sallee, <a href="http://dx.doi.org/10.1137/0605039">The middle-cut triangulations of the n-cube</a>, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407-419. MR0752044 (86c:05054). See Table 2.
%H A275518 John F. Sallee, <a href="http://dx.doi.org/10.1016/0012-365X(82)90190-X">A triangulation of the n-cube</a>, Discrete Math. 40 (1982), no. 1, 81-86. MR0676714 (84d:05065b)
%F A275518 a(n) = 1 + 2^(n-1) - n! + n!*Sum_{i=1..n} (2^(i-1)-1)/i!. - _Andrew Howroyd_, Sep 06 2023, after Maple program
%p A275518 p := proc(d,x)
%p A275518     add( x^i/i!,i=0..d) ;
%p A275518 end proc:
%p A275518 A275518 := proc(d)
%p A275518     d!*(p(d,2)/2-p(d,1))+2^(d-1)-d!/2+1 ;
%p A275518 end proc:
%p A275518 seq(A275518(d),d=1..18) ;
%t A275518 p[d_, x_] := Sum[x^i/i!, {i, 0, d}];
%t A275518 A275518[d_] := d!*(p[d, 2]/2 - p[d, 1]) + 2^(d - 1) - d!/2 + 1;
%t A275518 Table[A275518[d], {d, 1, 18}] (* _Jean-François Alcover_, Sep 06 2023, after Maple program *)
%o A275518 (PARI) a(n) = 1 + 2^(n-1) - n! + n!*sum(i=1, n, (2^(i-1)-1)/i!) \\ _Andrew Howroyd_, Sep 06 2023
%Y A275518 Cf. A239911, A019502, A019503, A019504, A166932.
%K A275518 nonn,easy
%O A275518 1,2
%A A275518 _R. J. Mathar_, Jul 31 2016
%E A275518 Terms a(19) and beyond from _Andrew Howroyd_, Sep 06 2023