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