A121634 -id:A121634 - cp's OEIS frontend

A121635 Number of deco polyominoes of height n, having no 2-cell columns starting at level 0. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

1, 1, 2, 8, 42, 264, 1920, 15840, 146160, 1491840, 16692480, 203212800, 2674425600, 37841126400, 572885913600, 9240898867200, 158228598528000, 2866422214656000, 54775863926784000, 1101208277385216000, 23234214178086912000, 513342323725271040000

Offset: 1

Emeric Deutsch, Aug 13 2006

			a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and the horizontal one has no 2-cell column starting at level 0.

Alois P. Heinz, Table of n, a(n) for n = 1..450 (n=2..101 from Muniru A Asiru)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Mark Dukes, Chris D White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes, Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.

Cf. A121634, A001710.

a:= n-> `if`(n=1, 1, (n^2-3*n+4)*(n-2)!/2): seq(a(n), n=1..23);

a(n) = A121634(n,0).
a(1)=1, a(n) = (n-2)!(n^2-3*n+4)/2 = A000142(n-2)*A152947(n) for n>=2.
a(1)=1, a(2)=1, a(n) = (n-2)*[(n-2)! + a(n-1)] for n>=3.
D-finite with recurrence a(n) +(-n-2)*a(n-1) +2*(n-1)*a(n-2) +2*(-n+4)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

Missing a(1) inserted by Alois P. Heinz, Nov 25 2018

A121636 Number of 2-cell columns starting at level 0 in all of deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Original entry on oeis.org

0, 1, 5, 23, 122, 754, 5364, 43308, 391824, 3929616, 43287840, 519711840, 6755460480, 94527008640, 1416783432960, 22646604153600, 384576130713600, 6914404440115200, 131217341055897600, 2621176954176614400

Offset: 1

Emeric Deutsch, Aug 13 2006

nonn

a(n)/(n-2)! is also the expected number of days it takes for the '100 Prisoners and a Light Bulb' to free themselves if there are n-1 prisoners if the prisoner on the first day is the counter for n>0. - Ron L.J. van den Burg, Jan 19 2020

			a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and only the vertical one has one 2-cell column starting at level 0.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Brett Ferry, 100 Prisoners and a Light Bulb Riddle & Solution, Math Hacks, 2015.

Cf. A121634, A000142.

a[1]:=0: a[2]:=1: for n from 3 to 23 do a[n]:=n*(n-2)!+(n-1)*a[n-1] od: seq(a[n],n=1..23);

a(1)=0, a(2)=1, a(n) = n(n-2)! + (n-1)*a(n-1) for n >= 3.
a(n) = Sum_{k=0..n-1} k*A121634(n,k).
a(n) = (n-1)!*(n^2-2n-1)/n + (n-1)!*(1/1 + 1/2 + ... + 1/n) (n >= 2). - Emeric Deutsch, Oct 22 2008
a(n) = (n-1)!*(h(n-1) + n - 2), n > 1, where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Oct 24 2010
a(n) = (n^2-3n+3)*(n-2)! + (n-1)*A000254(n-2), n > 2. - Ron L.J. van den Burg, Jan 19 2020
a(n+1) = (n-1)!*(n^2 + Sum_{k=1..n-1} k/(n-k)), n > 0. - Ron L.J. van den Burg, Jan 20 2020
Conjecture D-finite with recurrence a(n) +(-2*n+3)*a(n-1) +(n^2-5*n+7)*a(n-2) +(n-3)^2*a(n-3)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A121635 Number of deco polyominoes of height n, having no 2-cell columns starting at level 0. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

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