

A115241


Square array read by antidiagonals: T(n,p) is the number of linearly independent, homogeneous harmonic polynomials of degree n in p variables (n,p>=1).


3



1, 2, 0, 3, 2, 0, 4, 5, 2, 0, 5, 9, 7, 2, 0, 6, 14, 16, 9, 2, 0, 7, 20, 30, 25, 11, 2, 0, 8, 27, 50, 55, 36, 13, 2, 0, 9, 35, 77, 105, 91, 49, 15, 2, 0, 10, 44, 112, 182, 196, 140, 64, 17, 2, 0, 11, 54, 156, 294, 378, 336, 204, 81, 19, 2, 0, 12, 65, 210, 450, 672, 714, 540, 285, 100
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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(n1,x)+x*v(n1) and v(n,x)=u(n1,x)+v(n1,x)+1. See the Mathematica section at A209688. [From Clark Kimberling, Mar 12 2012]


REFERENCES

Harry Hochstadt, The Functions of Mathematical Physics, Wiley, New York (1971), p. 170; also Dover, New York (1986), p. 170.


LINKS

Table of n, a(n) for n=1..75.


FORMULA

T(n,p)=(2n+p2)binomial(n+p3,n1)/n (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^2y^2 and xy.
T(3,3)=7 because we can take, for example, x^33xy^2, x^33xz^2, y^33yx^2, y^33yz^2, z^33zx^2, z^33zy^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+p2)*binomial(n+p3, n1)/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


MATHEMATICA

T[n_, m_] := Binomial[n + m  3, n  1]*(2*n + m  2)/n a = Table[Table[T[n, m], {n, 1, m}], {m, 1, 12}] Flatten[a]


CROSSREFS

Cf. Diagonal terms are A097613; A209688.
Sequence in context: A284124 A187637 A141432 * A154559 A269133 A143324
Adjacent sequences: A115238 A115239 A115240 * A115242 A115243 A115244


KEYWORD

nonn,tabl


AUTHOR

Roger L. Bagula, Mar 04 2006


EXTENSIONS

Edited by N. J. A. Sloane, Mar 07 2006


STATUS

approved



