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%I A348210 #22 Feb 28 2024 01:36:03
%S A348210 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,
%T A348210 260,175,51,11,1,0,1,232,1085,981,369,76,13,1,0,1,603,4600,5719,2661,
%U A348210 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
%N A348210 Varma's Kosta numbers of semi-standard tableaux: array A(n>=2, k>=0) read by rising antidiagonals.
%C A348210 (More characteristic NAME desired.)
%C A348210 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).
%H A348210 G. C. Greubel, <a href="/A348210/b348210.txt">Antidiagonals n = 2..52, flattened</a>
%H A348210 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, 1997, Unpublished.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 03 2021]
%F A348210 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).
%F A348210 A(7,k) = 1 + 7*k*(k+1)*(11*k^2+11*k+8)/12.
%F A348210 A(8,k) = (2*k+1)*(4*k^2+6*k+3)*(4*k^2+2*k+1)/3.
%F A348210 A(9,k) = 1 + k*(k+1)*(289*k^4+578*k^3+581*k^2+292*k+108)/16.
%e A348210 The array starts in row n=2 with columns k>=0 as:
%e A348210 0 0 0 0 0 0 0 0 ...
%e A348210 1 1 1 1 1 1 1 1 ...
%e A348210 1 3 5 7 9 11 13 15 ...
%e A348210 1 6 16 31 51 76 106 141 ...
%e A348210 1 15 65 175 369 671 1105 1695 ...
%e A348210 1 36 260 981 2661 5916 11516 20385 ...
%e A348210 1 91 1085 5719 19929 54131 124501 254255 ...
%e A348210 Antidiagonal rows begin as:
%e A348210 0;
%e A348210 1, 0;
%e A348210 1, 1, 0;
%e A348210 1, 3, 1, 0;
%e A348210 1, 6, 5, 1, 0;
%e A348210 1, 15, 16, 7, 1, 0;
%e A348210 1, 36, 65, 31, 9, 1, 0;
%e A348210 1, 91, 260, 175, 51, 11, 1, 0;
%e A348210 1, 232, 1085, 981, 369, 76, 13, 1, 0;
%e A348210 1, 603, 4600, 5719, 2661, 671, 106, 15, 1, 0;
%p A348210 A348210 := proc(n,k)
%p A348210 local a,j ;
%p A348210 a := 0 ;
%p A348210 for j from 0 to floor((n-1)/2) do
%p A348210 a := a+ (-1)^j *binomial(n,j) *binomial( (n-2*j)*k+n-j-2,n-3) ;
%p A348210 end do:
%p A348210 -a/2 ;
%p A348210 end proc:
%p A348210 seq( seq( A348210(d-k,k),k=0..d-2),d=2..12) ;
%t A348210 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]}];
%t A348210 Table[A[n - k, k], {n, 2, 13}, {k, 0, n - 2}] // Flatten (* _Jean-François Alcover_, Mar 06 2023 *)
%o A348210 (Magma)
%o A348210 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)]]) >;
%o A348210 A348210:= func< n,k | A(n-k,k) >;
%o A348210 [A348210(n,k): k in [0..n-2], n in [2..13]]; // _G. C. Greubel_, Feb 28 2024
%o A348210 (SageMath)
%o A348210 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
%o A348210 def A348210(n,k): return A(n-k, k)
%o A348210 flatten([[A348210(n,k) for k in range(n-1)] for n in range(2,13)]) # _G. C. Greubel_, Feb 28 2024
%Y A348210 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.)
%K A348210 nonn,tabl,easy
%O A348210 2,8
%A A348210 _R. J. Mathar_, Oct 07 2021