A173020 Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows.

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

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

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

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).

Cf. A001764, A002293, A003408, A045943, A052149.

Sequence in context: A260285 A242499 A354622 * A334062 A157383 A232598

Adjacent sequences: A173017 A173018 A173019 * A173021 A173022 A173023

Keyword

easy,nonn,tabl

Author

R. J. Mathar, Nov 08 2010

Status

approved