Search: a121634 -id:a121634
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A121635
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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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+10
3
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1, 1, 2, 8, 42, 264, 1920, 15840, 146160, 1491840, 16692480, 203212800, 2674425600, 37841126400, 572885913600, 9240898867200, 158228598528000, 2866422214656000, 54775863926784000, 1101208277385216000, 23234214178086912000, 513342323725271040000
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OFFSET
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1,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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a:= n-> `if`(n=1, 1, (n^2-3*n+4)*(n-2)!/2): seq(a(n), n=1..23);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A121636
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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.
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+10
1
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0, 1, 5, 23, 122, 754, 5364, 43308, 391824, 3929616, 43287840, 519711840, 6755460480, 94527008640, 1416783432960, 22646604153600, 384576130713600, 6914404440115200, 131217341055897600, 2621176954176614400
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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