OFFSET
1,2
COMMENTS
A115241 is jointly generated with A209688 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+x*v(n-1) and v(n,x)=u(n-1,x)+v(n-1,x)+1. See the Mathematica section at A209688. - Clark Kimberling, Mar 12 2012
Clark Kimberling's recurrence in the previous comment yields an additional zero: 1, 0, 2, 0, 3, 2, ... - Georg Fischer, Sep 04 2021
REFERENCES
Harry Hochstadt, The Functions of Mathematical Physics, Wiley, New York (1971), p. 170; also Dover, New York (1986), p. 170.
FORMULA
T(n,p) = (2n+p-2) * binomial(n+p-3,n-1)/n for n>=1, p>=1.
EXAMPLE
T(1,1)=1 corresponds to the polynomial x.
T(n,1)=0 for n>=2 because no polynomial in x of degree >=2 is harmonic.
T(1,2)=2 because we can take, for example, x and y.
T(2,2)=2 because we can take, for example, x^2-y^2 and xy.
T(3,3)=7 because we can take, for example, x^3-3xy^2, x^3-3xz^2, y^3-3yx^2, y^3-3yz^2, z^3-3zx^2, z^3-3zy^2 and xyz.
The square array starts:
1, 2, 3, 4, 5, 6, 7, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 2, 7, 16, 30, 50, 77, ...
0, 2, 9, 25, 55, 105, 182, ...
0, 2, 11, 36, 91, 196, 378, ...
0, 2, 13, 49, 140, 336, 714, ...
0, 2, 15, 64, 204, 540, 1254, ...
...
MAPLE
T:=(n, p)->(2*n+p-2)*binomial(n+p-3, n-1)/n: for n from 1 to 10 do seq(T(n, p), p=1..10) od; # yields the 10 by 10 upper left corner of the square array
seq(seq(T(n, p+1-n), n=1..p), p=1..12); # yields the sequence; Georg Fischer, Sep 04 2021
MATHEMATICA
T[n_, m_] := Binomial[n + m - 3, n - 1]*(2*n + m - 2)/n; Table[Table[T[n, p+1-n], {n, 1, p}], {p, 1, 12}] // Flatten (* amended by Georg Fischer, Sep 04 2021 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 04 2006
EXTENSIONS
Edited by N. J. A. Sloane, Mar 07 2006
STATUS
approved