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Search: id:A115241
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| A115241 |
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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). |
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1
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| 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
(list; table; graph; listen)
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OFFSET
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1,2
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REFERENCES
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Harry Hochstadt, The Functions of Mathematical Physics, Wiley, New York (1971), p. 170; also Dover, New York (1986), p. 170.
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FORMULA
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T(n,p)=(2n+p-2)binomial(n+p-3,n-1)/n (n>=1,p>=1).
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EXAMPLE
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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...;
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MAPLE
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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
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MATHEMATICA
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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]
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CROSSREFS
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Diagonal terms are A097613
Sequence in context: A103489 A127479 A141432 this_sequence A154559 A143324 A097418
Adjacent sequences: A115238 A115239 A115240 this_sequence A115242 A115243 A115244
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KEYWORD
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nonn,tabl
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Mar 04 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 07 2006
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