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A177851
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Triangle read by rows: T(n, m) = binomial(n + m - 3, m - 1)*(2 * m + n - 2) / m, for n>=1 and 1<=m<=n.
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0
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1, 2, 2, 3, 5, 7, 4, 9, 16, 25, 5, 14, 30, 55, 91, 6, 20, 50, 105, 196, 336, 7, 27, 77, 182, 378, 714, 1254, 8, 35, 112, 294, 672, 1386, 2640, 4719, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875, 10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068
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OFFSET
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1,2
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COMMENTS
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This triangle sequence is the number of linearly independent homogeneous harmonic polynomials of degree m in n variables.
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REFERENCES
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Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 170
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LINKS
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FORMULA
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Row sums are ((3*n-1)*binomial(2*n-2,n)/(n-1)-1) for n>=2.
{1, 4, 15, 54, 195, 713, 2639, 9866, 37179, 140997,...}.
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EXAMPLE
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Triangle starts:
{1},
{2, 2},
{3, 5, 7},
{4, 9, 16, 25},
{5, 14, 30, 55, 91},
{6, 20, 50, 105, 196, 336},
{7, 27, 77, 182, 378, 714, 1254},
{8, 35, 112, 294, 672, 1386, 2640, 4719},
{9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875},
{10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068.
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MAPLE
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T := (n, m) -> ((2*m + n - 2)/m)*binomial(n + m - 3, m - 1):
for n from 1 to 10 do lprint(seq(T(n, k), k=1..n)) od; # Peter Luschny, Dec 16 2015
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MATHEMATICA
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Flatten[Table[Table[((2*m + n - 2)/m)*Binomial[n + m - 3, m - 1], {m, 1, n}], {n, 1, 10}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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