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A173020
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Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows.
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4
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1, 1, 3, 1, 9, 12, 1, 18, 66, 55, 1, 30, 210, 455, 273, 1, 45, 510, 2040, 3060, 1428, 1, 63, 1050, 6650, 17955, 20349, 7752, 1, 84, 1932, 17710, 74382, 148764, 134596, 43263, 1, 108, 3276, 40950, 245700, 753480, 1184040, 888030, 246675, 1, 135, 5220, 85260, 690606, 2992626, 7125300, 9161100, 5852925, 1430715
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OFFSET
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1,3
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COMMENTS
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REFERENCES
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Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53
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LINKS
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Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.
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FORMULA
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T(n, k) = R(n,k,3) with R(n,k,m)= binomial(n,k)*binomial(m*n,k-1)/n, 1<=k<=n.
T(n, 3) = n*(n-1)*(n-2)*(3*n-1)/4 = 3*A052149(n-1).
O.g.f. is series reversion with respect to x of x/((1+x)*(1+x*u)^3). - Peter Bala, Sep 12 2012
n-th row polynomial = x * hypergeom([1 - n, -3*n], [2], x). - Peter Bala, Aug 30 2023
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EXAMPLE
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The triangle starts in row n=1 as
1;
1, 3;
1, 9, 12;
1, 18, 66, 55;
1, 30, 210, 455, 273;
1, 45, 510, 2040, 3060, 1428;
1, 63, 1050, 6650, 17955, 20349, 7752;
1, 84, 1932, 17710, 74382, 148764, 134596, 43263;
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MATHEMATICA
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T[n_, k_, m_]:= Binomial[n, k]*Binomial[m*n, k-1]/n;
Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)
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PROG
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(Sage)
def A173020(n, k, m): return binomial(n, k)*binomial(m*n, k-1)/n
(Magma)
A173020:= func< n, k, m | Binomial(n, k)*Binomial(m*n, k-1)/n >;
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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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