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A173020 Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767.
Row sums are in A002293.
REFERENCES
Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53
LINKS
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 6.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.
FORMULA
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, n) = A001764(n).
T(n, n-1) = A003408(n-2).
T(n, 2) = A045943(n-1).
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
Sum_{k=1..n} T(n, k, 3) = binomial(4*n, n)/(3*n+1) = A002293(n). - G. C. Greubel, Feb 20 2021
n-th row polynomial = x * hypergeom([1 - n, -3*n], [2], x). - Peter Bala, Aug 30 2023
EXAMPLE
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;
MATHEMATICA
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 *)
PROG
(Sage)
def A173020(n, k, m): return binomial(n, k)*binomial(m*n, k-1)/n
flatten([[A173020(n, k, 3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 20 2021
(Magma)
A173020:= func< n, k, m | Binomial(n, k)*Binomial(m*n, k-1)/n >;
[A173020(n, k, 3): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 20 2021
CROSSREFS
Cf. A010054 (m=0), A001263 (m=1), A108767 (m=2), this sequence (m=3).
Sequence in context: A260285 A242499 A354622 * A334062 A157383 A232598
KEYWORD
easy,nonn,tabl
AUTHOR
R. J. Mathar, Nov 08 2010
STATUS
approved

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)