This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357367 #33 Aug 15 2025 10:34:08
%S A357367 1,0,2,0,6,12,0,24,120,120,0,120,1080,2520,1680,0,720,10080,40320,
%T A357367 60480,30240,0,5040,100800,604800,1512000,1663200,665280,0,40320,
%U A357367 1088640,9072000,33264000,59875200,51891840,17297280
%N A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.
%C A357367 T(n, k) is the cardinality of the set of all phylogenetic trees with linearly ordered children having n + 1 leaves and k internal vertices. (Proposition 4.16 in Deb and Sokal). - _Peter Luschny_, Aug 06 2025
%H A357367 Michael De Vlieger, <a href="/A357367/b357367.txt">Table of n, a(n) for n = 0..11475</a> (rows n = 0..150, flattened).
%H A357367 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 33.
%H A357367 Aleks Žigon Tankosič, <a href="https://arxiv.org/abs/2508.04754">Recurrence Relations for Some Integer Sequences Related to Ward Numbers</a>, arXiv:2508.04754 [math.CO], 2025. See pp. 3, 5.
%H A357367 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 12.
%F A357367 T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.
%F A357367 T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.
%F A357367 T(n, k) = (2*(n + k - 1))*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) with suitable boundary conditions (from Deb and Sokal). - _Peter Luschny_, Aug 06 2025
%e A357367 Triangle T(n, k) starts:
%e A357367 [0] 1;
%e A357367 [1] 0, 2;
%e A357367 [2] 0, 6, 12;
%e A357367 [3] 0, 24, 120, 120;
%e A357367 [4] 0, 120, 1080, 2520, 1680;
%e A357367 [5] 0, 720, 10080, 40320, 60480, 30240;
%e A357367 [6] 0, 5040, 100800, 604800, 1512000, 1663200, 665280;
%e A357367 [7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;
%p A357367 T := (n, k) -> add((-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!, m = 0..k):
%p A357367 seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
%p A357367 T := proc(n, k) option remember; if n = 0 and k = 0 then 1 elif k <= 0 or n < 0 then 0 else 2*(n + k - 1)*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) fi end:
%p A357367 for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Aug 06 2025
%t A357367 T[n_, k_] := Sum[(-1)^(m + k)*Binomial[n + k, n + m]*Binomial[n + m - 1, m - 1]*(n + m)!/m!, {m, 0, k}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 05 2025 *)
%o A357367 (SageMath)
%o A357367 def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)
%o A357367 def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)
%o A357367 for m in range(k + 1)))
%o A357367 for n in range(8): print([T(n, k) for k in range(n+1)])
%Y A357367 Cf. A032037 (row sums), A271703, A386789.
%K A357367 nonn,tabl
%O A357367 0,3
%A A357367 _Peter Luschny_, Sep 26 2022
%E A357367 New name using a formula of Deb and Sokal by _Peter Luschny_, Aug 06 2025