VOOZH about

URL: https://oeis.org/A094796

⇱ A094796 - OEIS

login
A094796
Triangle read by rows giving coefficients of polynomials arising in successive differences of central binomial numbers.
3
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.
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
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