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

Author

R. J. Mathar, Nov 08 2010

Keywords

• easy
• nonn
• tabl

Comments

The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767.

Row sums are in A002293.

Examples

			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;
		

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.

Crossrefs

Cf. A010054 (m=0), A001263 (m=1), A108767 (m=2), this sequence (m=3).

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

Programs

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

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 *)

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

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