A348211 Triangle read by rows giving coefficients of polynomials arising as numerators of certain Hilbert series.

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, 2997, 225, 1, 1, 595, 15049, 62755, 62755, 15049, 595, 1, 1, 1576, 72496, 527911, 985758, 527911, 72496, 1576, 1, 1, 4203, 341166, 4094762, 12956604, 12956604, 4094762, 341166, 4203, 1

Offset: 3

Author

R. J. Mathar, Oct 07 2021

Keywords

• tabl
• nonn

Comments

This corrects 544 -> 554 in row 8 of A013561.

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.

Examples

			Triangle begins:
  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,   2997,    225,     1;
  1,  595, 15049,  62755,  62755,  15049,   595,    1;
  1, 1576, 72496, 527911, 985758, 527911, 72496, 1576,   1;
		

Links

• G. C. Greubel, Rows n = 3..53 of the triangle, flattened
• D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]

Crossrefs

Cf. A012249 (row sums), A013561, A013630.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
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
p:= func< n,x | (1-x)^(n-2)*(&+[A(n,k)*x^k: k in [0..n]]) >;
A348211:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
[A348211(n,k): k in [0..n-3], n in [3..15]]; // G. C. Greubel, Feb 28 2024

Maple

read("transforms"):
A348211_row := proc(n)
    local x,b,opoly ;
    opoly := n-2 ;
    [seq(A348210(n,k),k=0..opoly-1)] ;
    b := BINOMIALi(%) ;
    add( op(i,b)*x^(i-1)*(1-x)^(opoly-i),i=1..nops(b)) ;
    seq( coeff(%,x,i),i=0..opoly-1) ;
end proc:
for n from 3 to 12 do
    print(A348211_row(n)) ;
end do: # R. J. Mathar, Oct 10 2021

Mathematica

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]}];
row[n_] := Switch[n, 3, {1}, 4, {1, 1}, _, FindGeneratingFunction[Table[A348210[n, k], {k, 0, n-2}], x] // Numerator // CoefficientList[#, x]& // Abs];
Table[row[n], {n, 3, 12}] // Flatten (* Jean-François Alcover, Apr 23 2023 *)

SageMath

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
def p(n,x): return (1-x)^(n-2)*sum( A(n,k)*x^k for k in range(n+1) )
def A348211(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
flatten([[A348211(n,k) for k in range(n-2)] for n in range(3,17)]) # G. C. Greubel, Feb 28 2024

Formula

Sum_{k=0..n-3} T(n, k) = A012249(n-2) (row sums).

From G. C. Greubel, Feb 28 2024: (Start)

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.

T(n, n-k) = T(n, k). (End)