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A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces.

%I A281620 #46 Apr 06 2020 15:57:46
%S A281620 1,7,1,67,13,1,821,181,21,1,12281,2906,406,31,1,217015,53719,8359,799,
%T A281620 43,1,4424071,1129899,188707,20637,1429,57,1,102207817,26710345,
%U A281620 4690249,561481,45385,2377,73,1,2639010709,701908264,127951984,16349374,1469026,91216,3736,91,1
%N A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces.
%H A281620 So Okada, <a href="https://arxiv.org/abs/0910.2014">Homological mirror symmetry of Fermat polynomials</a>, arxiv:0910.2014 [math.AG], 2009-2010.
%F A281620 The formula given by Okada needs to be corrected as follows:
%F A281620 Sum_{j=0..n-1} Sum_{i=0..n-1-j} n^j * binomial(n,j) * (-1)^(i+n+j) * binomial(n-2-j+1,i+1) * q^i.
%F A281620 From _Peter Luschny_, Jan 26 2017: (Start)
%F A281620 T(n,k) = [x^k] Sum_{j=0..n-1} t(j, n) for n>=2 and 0<=k<=n-2 with t(j,n) = (-1)^(j+n)*binomial(n,j)*(1-(1-x)^(n-1-j))*x^(-1)*n^j.
%F A281620 T(n,k) = [x^k] ((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x). (End)
%e A281620 The first few polynomials are 1; q + 7; q^2 + 13*q + 67; ...
%e A281620 Triangle begins:
%e A281620         1;
%e A281620         7,       1;
%e A281620        67,      13,      1;
%e A281620       821,     181,     21,     1;
%e A281620     12281,    2906,    406,    31,    1;
%e A281620    217015,   53719,   8359,   799,   43,  1;
%e A281620   4424071, 1129899, 188707, 20637, 1429, 57, 1;
%e A281620   ...
%p A281620 T:= n-> (p-> seq(coeff(p, q, i), i=0..n-2))(add(add(n^j*
%p A281620          binomial(n, j)*(-1)^(i+n+j)*binomial(n-2-j+1, i+1)*
%p A281620          q^i, i=0..n-1-j), j=0..n-1)):
%p A281620 seq(T(n), n=2..10);  # _Alois P. Heinz_, Jan 25 2017
%p A281620 # Alternatively:
%p A281620 t := n -> factor(((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x)):
%p A281620 seq(seq(coeff(t(n),x,k),k=0..n-2),n=2..10); # _Peter Luschny_, Jan 26 2017
%t A281620 T[n_] := ((-n-x+1)^n+(x-1)(1-n)^n-(-n)^n x) (-1)^n/((x-1) x); Table[CoefficientList[T[n],x],{n,2,10}] // Flatten (* _Peter Luschny_, Jan 26 2017 *)
%o A281620 (Sage)
%o A281620 def fermat(n):
%o A281620     q = polygen(ZZ, 'q')
%o A281620     return sum(n**j * binomial(n, j) * (-1)**(i + n + j) *
%o A281620                binomial(n - 2 - j + 1, i + 1) * q**i
%o A281620                for j in range(n - 1)
%o A281620                for i in range(n - 1 - j))
%o A281620 (Sage) # Alternatively:
%o A281620 def A281620_row(n):
%o A281620     x = polygen(ZZ, 'x')
%o A281620     p = (((-n-x+1)^n + (x-1)*(1-n)^n - (-n)^n*x)*(-1)^n)//((x-1)*x)
%o A281620     return p.list()
%o A281620 for n in (2..10): print(A281620_row(n)) # _Peter Luschny_, Jan 26 2017
%Y A281620 Row sums give A007778(n-1), alternating row sums are A281596.
%K A281620 nonn,tabl
%O A281620 2,2
%A A281620 _F. Chapoton_, Jan 25 2017