A217922 Triangle read by rows: labeled trees counted by improper edges.
1, 1, 2, 1, 6, 7, 3, 24, 46, 40, 15, 120, 326, 430, 315, 105, 720, 2556, 4536, 4900, 3150, 945, 5040, 22212, 49644, 70588, 66150, 38115, 10395, 40320, 212976, 574848, 1011500, 1235080, 1032570, 540540, 135135
Offset: 1
Examples
Triangle begins: \ k 0....1....2....3....4...... n 1 |..1 2 |..1 3 |..2....1 4 |..6....7....3 5 |.24...46...40....15 6 |120..326..430...315...105 T(4,2) = 3 because we have 1->3->4->2, 1->4->2->3, 1->4->3->2, in each of which the last 2 edges are improper.
Links
- G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened
- J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
- William Y. C. Chen, Amy M. Fu, and Elena L. Wang, A Grammatical Calculus for the Ramanujan Polynomials, arXiv:2506.01649 [math.CO], 2025. See p. 3.
- Dominique Dumont and Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 17).
- Matthieu Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
- Lucas Randazzo, Arboretum for a generalization of Ramanujan polynomials, arXiv:1905.02083 [math.CO], 2019.
- Jiang Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan Journal 3 (1999) 1, 45-54, [DOI]
Programs
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Magma
function T(n,k) // T = A217922 if k lt 0 or k gt n-2 then return 0; elif k eq 0 then return Factorial(n-1); else return (n-1)*T(n-1,k) + (n+k-3)*T(n-1,k-1); end if; end function; [1] cat [T(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, Jan 10 2025
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Mathematica
T[n_, k_]:= T[n,k]= If[k<0 || k>n-2, 0, If[k==0, (n-1)!, (n-1)*T[n-1,k] + (n+k-3)*T[n-1, k-1]]]; Join[{1}, Table[T[n,k], {n,12}, {k,0,n-2}]//Flatten] (* modified by G. C. Greubel, May 07 2019 *) -
SageMath
def T(n, k): if k==0: return factorial(n-1) elif (k<0 or k > n-2): return 0 else: return (n-1)*T(n-1, k) + (n+k-3)* T(n-1, k-1) flatten([1] + [[T(n, k) for k in (0..n-2)] for n in (2..12)]) # G. C. Greubel, May 07 2019
Formula
T(n, k) = (n-1)*T(n-1, k) + (n+k-3)*T(n-1, k-1), for 1 <= k <= n-2, with T(n, 0) = (n-1)!. - G. C. Greubel, Jan 10 2025
Comments