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A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

%I A133458 #23 Sep 08 2022 08:45:31
%S A133458 1,35,393,2128,8135,24017,60691,134512,273127,512365,908755,1528688,
%T A133458 2473325,3852919,5832765,8582336,12354469,17395119,24072133,32726960,
%U A133458 43874139,57971221,75715487,97702640,124853275,157924585,198105727
%N A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.
%C A133458 The middle coefficients for dimension d>=1 are in A000012, A000027, A077043, A005900, A077044, A071816, here, the d-th row in A077042.
%C A133458 For d=8 the sequence starts 1, 70, 1107, 8092, 38165, 135954, 398567, 1012664, 2306025, ... and for d=9 it starts 1, 126, 3139, 30276, 180325, 767394, 2636263, 7635987, 19610233, ... - _R. J. Mathar_, Sep 04 2011
%H A133458 Vincenzo Librandi, <a href="/A133458/b133458.txt">Table of n, a(n) for n = 1..10000</a>
%H A133458 R. P. Stanley, <a href="http://dx.doi.org/10.1137/0601021">Weyl groups, the Hard Lefschetz Theorem and the Sperner property</a>, SIAM J. Alg. Disc. Meth. 1 (2) (1980) 168, see eq. (4).
%F A133458 From _R. J. Mathar_, Feb 19 2010: (Start)
%F A133458 a(n)= 2*a(n-1) +4*a(n-2) -10*a(n-3) -5*a(n-4) +20*a(n-5) -20*a(n-7) +5*a(n-8) +10*a(n-9) -4*a(n-10) -2*a(n-11) +a(n-12).
%F A133458 G.f.: x*(1+33*x +319*x^2 +1212*x^3 +2662*x^4 +3320*x^5 +2662*x^6 +1212*x^7 +319*x^8 +33*x^9 +x^10)/ ((1+x)^5 * (1-x)^7).
%F A133458 a(n) = -25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536. (End)
%p A133458 f:=(L,d)->(sum(x^k,k=0..L-1))^d; A:=[seq(coeff(f(j,7),x,floor(7*(j-1)/2)),j=1..25)];
%p A133458 A133458 := proc(n) -25/512*(-1)^n +2261/23040*n^2 -91/1536*(-1)^n*n^2 -77/1536*(-1)^n*n^4 +847/4608*n^4 +5887/11520*n^6 +25/512 ; end proc: # _R. J. Mathar_, Sep 05 2011
%o A133458 (Magma) [-25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536 : n in [1..40]]; // _Vincenzo Librandi_, Sep 07 2011
%Y A133458 Cf. A077043, A005900, A077044, A071816.
%K A133458 nonn
%O A133458 1,2
%A A133458 Leonid Chindelevitch (leonidus(AT)mit.edu), Dec 22 2007
%E A133458 More terms from _R. J. Mathar_, Feb 19 2010