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

1, 3, 1, 9, 15, 6, 27, 108, 135, 42, 81, 594, 1539, 1530, 456, 243, 2835, 12555, 25245, 22122, 6120, 729, 12393, 83835, 281475, 482436, 383292, 101520, 2187, 51030, 489888, 2466450, 6916833, 10546200, 7786692, 1980720

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..35.

Formula

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

Example

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.
From M. F. Hasler, Nov 15 2019: (Start)
The table reads:
  n  |  row(n)
  0  |    1
  1  |    3      1
  2  |    9     15       6
  3  |   27    108     135       42
  4  |   81    594    1539     1530      456
  5  |  243   2835   12555    25245    22122      6120
  6  |  729  12393   83835   281475   482436    383292    101520
  7  | 2187  51030  489888  2466450  6916833  10546200   7786692   1980720
  8  | 6561 201204 2602530 18329976 75981969 186899076 260520300 181218384 44634240
(End)

Maple

Dnk := proc(n, k)
    option remember;
    if k < 0 then
        0 ;
    elif k = 0 then
        binomial(2*n, n) ;
    else
        procname(n+1, k-1)-procname(n, k-1) ;
    end if;
end proc:
A094796 := proc(n, k)
    local xyvec, i, x ;
    xyvec := [] ;
    for i from 0 to n do
        xyvec := [op(xyvec), [i, Dnk(i, n)*mul(i+j, j=1..n)/Dnk(i, 0)]] ;
    end do:
    CurveFitting[PolynomialInterpolation](xyvec, x) ;
    coeff(%, x, n-k) ;
end proc: # R. J. Mathar, Nov 19 2019

Mathematica

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[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]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 01 2024, after R. J. Mathar *)

Prog

(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

Crossrefs

Cf. A000984 (central binomial coefficients), A163771 (square array of central binomial coefficients and higher differences), A000244 (column k=0).

Main diagonal gives A098461.

Sequence in context: A157399 A288852 A162749 * A056843 A076806 A111568

Adjacent sequences: A094793 A094794 A094795 * A094797 A094798 A094799

Keyword

nonn,tabl

Author

Benoit Cloitre, Jun 11 2004

Extensions

Corrected and edited by M. F. Hasler, following observations by R. J. Mathar and Don Reble, Nov 15 2019

More terms from Don Reble, Nov 15 2019

Status

approved