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A121586 Number of columns in all 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. 3
1, 1, 3, 13, 70, 446, 3276, 27252, 253296, 2602224, 29288160, 358457760, 4740577920, 67375532160, 1024208720640, 16583626886400, 284953145702400, 5178968115148800, 99268112350310400, 2001336861359001600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is also the largest entry in the cycle containing 1, summed over all permutations of {1,2,...,n}. Example: a(3) = 13 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132), written in cycle notation, yield 1+1+2+3+3+3=13. - Emeric Deutsch, Nov 10 2008

LINKS

Table of n, a(n) for n=0..19.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

FORMULA

a(n) = (n+1)! - |s(n+1,2)|, where s(n,k) are the signed Stirling numbers of the first kind (A008275).

Recurrence relation: a(n) = n*a(n-1) + (n-1)!*(n-1); (see the Barcucci et al. reference, p. 34).

a(n) = Sum_{k=1..n} k*A094638(n,k).

From Emeric Deutsch, Nov 10 2008: (Start)

a(n) = (n-1)!*(n^2 + n - 1 - n*H(n-1)) for n >= 1, where H(j) = 1/1+1/2+...+1/j.

a(n) = Sum_{k=1..n} k*A145888(n,k) for n >= 1. (End)

From Gary Detlefs, Sep 12 2010: (Start)

a(n) = n!*((n+1) - h(n)), where h(n) = Sum_{k=1..n} 1/k.

a(n) = (n+1)! - A000254(n). (End)

E.g.f.: (1 - (x - 1)*log(1 - x))/(x - 1)^2. - Benedict W. J. Irwin, Sep 27 2016

EXAMPLE

a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 1 and 2 columns.

MAPLE

a[0] := 1; a[1] := 1: for n from 2 to 22 do a[n] := n*a[n-1] + (n-1)!*(n-1) od:

seq(a[n], n = 0..22);

# Second program:

egf := (1 - (x - 1)*log(1 - x))/(x - 1)^2: ser := series(egf, x, 20):

seq(n!*coeff(ser, x, n), n = 0..19); # Peter Luschny, Dec 09 2021

MATHEMATICA

Join[{1}, Table[CoefficientList[Series[((x-1)Log[1-x]-x-1)/(x-1)^3, {x, 0, 20}], x][[n]] (n-1)!, {n, 1, 20}]] (* Benedict W. J. Irwin, Sep 27 2016 *)

CROSSREFS

Cf. A008275, A094638, A145888.

Sequence in context: A274379 A192209 A154677 * A024337 A001495 A284217

Adjacent sequences:  A121583 A121584 A121585 * A121587 A121588 A121589

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 14 2006

EXTENSIONS

a(0) = 1 prepended by Peter Luschny, Dec 09 2021

STATUS

approved

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Last modified October 2 05:15 EDT 2022. Contains 357191 sequences. (Running on oeis4.)