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%I #18 Feb 29 2024 01:45:04
%S 1,1,1,1,3,1,1,11,11,1,1,31,90,31,1,1,85,554,554,85,1,1,225,2997,6559,
%T 2997,225,1,1,595,15049,62755,62755,15049,595,1,1,1576,72496,527911,
%U 985758,527911,72496,1576,1,1,4203,341166,4094762,12956604,12956604,4094762,341166,4203,1
%N Triangle read by rows giving coefficients of polynomials arising as numerators of certain Hilbert series.
%C This corrects 544 -> 554 in row 8 of A013561.
%C Write the g.f. of row n of A348210 as a rational polynomial nu(x)/(1-x)^(n-2). The triangle contains the coefficients [x^k] nu(x) in row n.
%H G. C. Greubel, <a href="/A348211/b348211.txt">Rows n = 3..53 of the triangle, flattened</a>
%H D.-N. Verma, <a href="/A012249/a012249.pdf">Towards Classifying Finite Point-Set Configurations</a>, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]
%F Sum_{k=0..n-3} T(n, k) = A012249(n-2) (row sums).
%F From _G. C. Greubel_, Feb 28 2024: (Start)
%F T(n, k) = [x^k]( (1-x)^(n-2) * Sum_{k=0..n-3} A(n,k)*x^k ), where A(n,k) is the array of A348210.
%F T(n, n-k) = T(n, k). (End)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 11, 11, 1;
%e 1, 31, 90, 31, 1;
%e 1, 85, 554, 554, 85, 1;
%e 1, 225, 2997, 6559, 2997, 225, 1;
%e 1, 595, 15049, 62755, 62755, 15049, 595, 1;
%e 1, 1576, 72496, 527911, 985758, 527911, 72496, 1576, 1;
%p read("transforms"):
%p A348211_row := proc(n)
%p local x,b,opoly ;
%p opoly := n-2 ;
%p [seq(A348210(n,k),k=0..opoly-1)] ;
%p b := BINOMIALi(%) ;
%p add( op(i,b)*x^(i-1)*(1-x)^(opoly-i),i=1..nops(b)) ;
%p seq( coeff(%,x,i),i=0..opoly-1) ;
%p end proc:
%p for n from 3 to 12 do
%p print(A348211_row(n)) ;
%p end do: # _R. J. Mathar_, Oct 10 2021
%t A348210[n_, k_] := (-1/2)*Sum[(-1)^j*Binomial[n, j]* Binomial[(n-2*j)*k+n-j-2, n-3], {j, 0, Floor[(n-1)/2]}];
%t row[n_] := Switch[n, 3, {1}, 4, {1, 1}, _, FindGeneratingFunction[Table[A348210[n, k], {k, 0, n-2}], x] // Numerator // CoefficientList[#, x]& // Abs];
%t Table[row[n], {n, 3, 12}] // Flatten (* _Jean-François Alcover_, Apr 23 2023 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 50);
%o A:= func< n, k | (&+[(-1)^(j+1)*Binomial(n, j)*Binomial((n-2*j)*k+n-j-2, n-3)/2 : j in [0..Floor((n-1)/2)]]) >; // A=A348210
%o p:= func< n,x | (1-x)^(n-2)*(&+[A(n,k)*x^k: k in [0..n]]) >;
%o A348211:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
%o [A348211(n,k): k in [0..n-3], n in [3..15]]; // _G. C. Greubel_, Feb 28 2024
%o (SageMath)
%o def A(n, k): return sum( (-1)^(j+1)*binomial(n, j)*binomial((n-2*j)*k+n-j-2, n-3) for j in range(1+(n-1)//2) )/2 # A = A348210
%o def p(n,x): return (1-x)^(n-2)*sum( A(n,k)*x^k for k in range(n+1) )
%o def A348211(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
%o flatten([[A348211(n,k) for k in range(n-2)] for n in range(3,17)]) # _G. C. Greubel_, Feb 28 2024
%Y Cf. A012249 (row sums), A013561, A013630.
%K tabl,nonn
%O 3,5
%A _R. J. Mathar_, Oct 07 2021
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