OFFSET
3,5
COMMENTS
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D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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
KEYWORD
tabl,nonn
AUTHOR
R. J. Mathar, Oct 07 2021
STATUS
approved