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

3,5

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.

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D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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]

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)

Example

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;

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 *)

Prog

(Magma)
R<x>:=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
(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

Crossrefs

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

Sequence in context: A256895 A223256 A013561 * A176468 A176421 A168552

Adjacent sequences: A348208 A348209 A348210 * A348212 A348213 A348214

Keyword

tabl,nonn

Author

R. J. Mathar, Oct 07 2021

Status

approved