A348210 Varma's Kosta numbers of semi-standard tableaux: array A(n>=2, k>=0) read by rising antidiagonals.

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 15, 16, 7, 1, 0, 1, 36, 65, 31, 9, 1, 0, 1, 91, 260, 175, 51, 11, 1, 0, 1, 232, 1085, 981, 369, 76, 13, 1, 0, 1, 603, 4600, 5719, 2661, 671, 106, 15, 1, 0, 1, 1585, 19845, 33922, 19929, 5916, 1105, 141, 17, 1, 0, 1, 4213, 86725, 204687, 151936, 54131, 11516, 1695, 181, 19, 1, 0

Offset: 2

Author

R. J. Mathar, Oct 07 2021

Keywords

• nonn
• tabl
• easy

Comments

(More characteristic NAME desired.)

Each row is a polynomial in k, which implies that the inverse binomial transformation of each row is a finite sequence and that the row can be represented by a rational generating function (A348211).

Examples

			The array starts in row n=2 with columns k>=0 as:
  0   0    0    0     0     0      0      0 ...
  1   1    1    1     1     1      1      1 ...
  1   3    5    7     9    11     13     15 ...
  1   6   16   31    51    76    106    141 ...
  1  15   65  175   369   671   1105   1695 ...
  1  36  260  981  2661  5916  11516  20385 ...
  1  91 1085 5719 19929 54131 124501 254255 ...
Antidiagonal rows begin as:
  0;
  1,   0;
  1,   1,    0;
  1,   3,    1,    0;
  1,   6,    5,    1,    0;
  1,  15,   16,    7,    1,    0;
  1,  36,   65,   31,    9,    1,   0;
  1,  91,  260,  175,   51,   11,   1,   0;
  1, 232, 1085,  981,  369,   76,  13,   1,  0;
  1, 603, 4600, 5719, 2661,  671, 106,  15,  1,  0;
		

Links

• G. C. Greubel, Antidiagonals n = 2..52, 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• D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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 03 2021]

Crossrefs

Cf. A005043 (column k=1), A007043 (k=2), A264608 (k=3), A272393 (k=4), A005408 (row n=4), A005891 (n=5), A005917 (n=6), A348211 (condensed g.f.)

Programs

Magma

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)]]) >;
A348210:= func< n,k | A(n-k,k) >;
[A348210(n,k): k in [0..n-2], n in [2..13]]; // G. C. Greubel, Feb 28 2024

Maple

A348210 := proc(n,k)
    local a,j ;
    a := 0 ;
    for j from 0 to floor((n-1)/2) do
            a := a+ (-1)^j *binomial(n,j) *binomial( (n-2*j)*k+n-j-2,n-3) ;
    end do:
    -a/2 ;
end proc:
seq( seq( A348210(d-k,k),k=0..d-2),d=2..12) ;

Mathematica

A[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]}];
Table[A[n - k, k], {n, 2, 13}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Mar 06 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
def A348210(n,k): return A(n-k, k)
flatten([[A348210(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, Feb 28 2024

Formula

A(n,k) = (-1/2)*Sum_{j=0..floor((n-1)/2)} (-1)^j *binomial(n,j) *binomial((n-2*j)*k+n-j-2,n-3).

A(7,k) = 1 + 7*k*(k+1)*(11*k^2+11*k+8)/12.

A(8,k) = (2*k+1)*(4*k^2+6*k+3)*(4*k^2+2*k+1)/3.

A(9,k) = 1 + k*(k+1)*(289*k^4+578*k^3+581*k^2+292*k+108)/16.