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A094796 Triangle read by rows giving coefficients of polynomials arising in successive differences of central binomial numbers.

%I A094796 #28 Apr 01 2024 13:02:22
%S A094796 1,3,1,9,15,6,27,108,135,42,81,594,1539,1530,456,243,2835,12555,25245,
%T A094796 22122,6120,729,12393,83835,281475,482436,383292,101520,2187,51030,
%U A094796 489888,2466450,6916833,10546200,7786692,1980720
%N A094796 Triangle read by rows giving coefficients of polynomials arising in successive differences of central binomial numbers.
%C A094796 Let D_0(n)=binomial(2*n,n) and D_{k+1}(n)=D_{k}(n+1)-D_{k}(n); then D_{k}(n)*(n+1)*(n+2)*...*(n+k) = binomial(2*n,n)*P_{k}(n) where P_{k} is a polynomial with integer coefficients of degree k.
%F A094796 T(n,0) = 3^n. T(n,1) = A027472(n+2) + 6*A027472(n+1). T(n,2) = 3*(2*A036217(n-2) + 15*A036217(n-3) + 18*A036217(n-4)). - _R. J. Mathar_, Nov 19 2019
%e A094796 The third differences of the central binomial numbers are given by D_3(n) = binomial(2*n,n)*(n+1)*(n+2)*(n+3)*(27*n^3 + 108*n^2 + 135*n + 42) and the fourth row of the triangle is 27, 108, 135, 42.
%e A094796 From _M. F. Hasler_, Nov 15 2019: (Start)
%e A094796 The table reads:
%e A094796   n  |  row(n)
%e A094796   0  |    1
%e A094796   1  |    3      1
%e A094796   2  |    9     15       6
%e A094796   3  |   27    108     135       42
%e A094796   4  |   81    594    1539     1530      456
%e A094796   5  |  243   2835   12555    25245    22122      6120
%e A094796   6  |  729  12393   83835   281475   482436    383292    101520
%e A094796   7  | 2187  51030  489888  2466450  6916833  10546200   7786692   1980720
%e A094796   8  | 6561 201204 2602530 18329976 75981969 186899076 260520300 181218384 44634240
%e A094796 (End)
%p A094796 Dnk := proc(n,k)
%p A094796     option remember;
%p A094796     if k < 0 then
%p A094796         0 ;
%p A094796     elif k = 0 then
%p A094796         binomial(2*n,n) ;
%p A094796     else
%p A094796         procname(n+1,k-1)-procname(n,k-1) ;
%p A094796     end if;
%p A094796 end proc:
%p A094796 A094796 := proc(n,k)
%p A094796     local xyvec,i,x ;
%p A094796     xyvec := [] ;
%p A094796     for i from 0 to n do
%p A094796         xyvec := [op(xyvec),[i,Dnk(i,n)*mul(i+j,j=1..n)/Dnk(i,0)]] ;
%p A094796     end do:
%p A094796     CurveFitting[PolynomialInterpolation](xyvec,x) ;
%p A094796     coeff(%,x,n-k) ;
%p A094796 end proc: # _R. J. Mathar_, Nov 19 2019
%t A094796 Dnk[n_, k_] := Dnk[n, k] = Which[k < 0, 0, k == 0, Binomial[2*n, n], True, Dnk[n + 1, k - 1] - Dnk[n, k - 1]];
%t A094796 T[n_, k_] := Module[{xyvec, i, x , ip}, xyvec = {}; For[i = 0, i <= n, i++, AppendTo[xyvec, {i, Dnk[i, n]*Product[i + j, {j, 1, n}]/Dnk[i, 0]}]]; ip = InterpolatingPolynomial[xyvec, x]; Coefficient[ip, x, n - k]];
%t A094796 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 01 2024, after _R. J. Mathar_ *)
%o A094796 (PARI) apply( {A094796_row(n,D(n,k)=if(k,D(n+1,k-1)-D(n,k-1),binomial(2*n,n)))=Vec(polinterpolate([0..n],vector(n+1,k,D(k--,n)*(n+k)!/k!/binomial(2*k,k))))}, [0..8]) \\ _M. F. Hasler_, Nov 15 2019
%Y A094796 Cf. A000984 (central binomial coefficients), A163771 (square array of central binomial coefficients and higher differences), A000244 (column k=0).
%Y A094796 Main diagonal gives A098461.
%K A094796 nonn,tabl
%O A094796 0,2
%A A094796 _Benoit Cloitre_, Jun 11 2004
%E A094796 Corrected and edited by _M. F. Hasler_, following observations by _R. J. Mathar_ and _Don Reble_, Nov 15 2019
%E A094796 More terms from _Don Reble_, Nov 15 2019