A308679 Kuba-Panholzer Table 2 pattern 123, 321 for Stirling permutation k = 2.

5, 33, 180, 919, 4560, 22332, 108733, 528298, 2566516, 12480403, 60784064, 296593256, 1450124169, 7104618375, 34878823088, 171572357252, 845605268800, 4175311417840, 20652607880698, 102326794307846, 507804406403540, 2523838310290891, 12561785900116608, 62608677333571728

Offset

3,1

Links

Table of n, a(n) for n=3..26.

Markus Kuba, Alois Panholzer, Stirling permutations containing a single pattern of length three, Australasian Journal of Combinatorics (2019) Vol. 74, No. 2, 216-239.

Formula

For k = 2, a(n) = (Sum_{j = 0..k} C(k, j) * (-1)^j * ((k-1) * (Sum_{m = 0..(n-j)} C(n-j-1+(k-1)*m, n-j-m) * (2*C(n-j+2, m)/(n-j+2) - C(n-j+1, m)/(n-j+1))) + (Sum_{m = 0..(n+1-j)} C(n-j+(k-1)*m, n-j+1-m) * (2*C(n+1-j+2, m)/(n+1-j+2) - C(n+1-j+1, m)/(n+1-j+1))))) + (2*(2-k)/(n+1)) * (Sum_{m = 0..(n-1)} C(n+1, m) * C(n-2+(k-1)*m, n-1-m)) - (4/(n+2)) * (Sum_{m = 0..n} C(n+2, m) * C(n-1+(k-1)*m, n-m)) + (Sum_{m = 0..n} C(n, m) * C(n+(k-1)*m-1, n-m)/(n+1-m)) (See Kuba-Panholzer paper).

Mathematica

With[{k = 2}, Table[Sum[Binomial[k, j] (-1)^j*((k - 1) Sum[Binomial[n - j - 1 + (k - 1) m, n - j - m] (2 Binomial[#, m]/# &[n - j + 2] - Binomial[#, m]/# &[n - j + 1]), {m, 0, n - j}] + Sum[Binomial[n - j + (k - 1) m, n - j + 1 - m] (2 Binomial[#, m]/# &[n + 1 - j + 2] - Binomial[#, m]/# &[n + 1 - j + 1]), {m, 0, n + 1 - j}]), {j, 0, k}] + (2 (2 - k))/(n + 1) Sum[Binomial[n + 1, m] Binomial[n - 2 + (k - 1) m, n - 1 - m], {m, 0, n - 1}] - (4/(n + 2)) Sum[Binomial[n + 2, m] Binomial[n - 1 + (k - 1) m, n - m], {m, 0, n}] + Sum[Binomial[n, m] Binomial[n + (k - 1) m - 1, n - m]/(n + 1 - m), {m, 0, n}], {n, 0, 25}]]

Prog

(PARI) a(n) = my(k=2, C=binomial); sum(j = 0, k, C(k, j) * (-1)^j * ((k-1) * sum(m = 0, (n-j), C(n-j-1+(k-1)*m, n-j-m) * (2*C(n-j+2, m)/(n-j+2) - C(n-j+1, m)/(n-j+1))) + sum(m = 0, (n+1-j), C(n-j+(k-1)*m, n-j+1-m) * (2*C(n+1-j+2, m)/(n+1-j+2) - C(n+1-j+1, m)/(n+1-j+1))))) + (2*(2-k)/(n+1)) * sum(m = 0, (n-1), C(n+1, m) * C(n-2+(k-1)*m, n-1-m)) - (4/(n+2)) * sum(m = 0, n, C(n+2, m) * C(n-1+(k-1)*m, n-m)) + sum(m = 0, n, C(n, m) * C(n+(k-1)*m-1, n-m)/(n+1-m)) \\ Jason Yuen, May 28 2025

Crossrefs

Cf. A001700, A003517.

Sequence in context: A270690 A273141 A270726 * A272833 A050915 A091056

Adjacent sequences: A308676 A308677 A308678 * A308680 A308681 A308682

Keyword

nonn,easy

Author

Michael De Vlieger, Jun 16 2019

Status

approved