cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A121580 Number of cells in column 1 of 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.

Original entry on oeis.org

1, 3, 11, 53, 317, 2237, 18077, 164237, 1656077, 18348557, 221561357, 2895986957, 40737113357, 613623026957, 9854521894157, 168083120422157, 3034505335078157, 57810369261862157, 1159018646647078157
Offset: 1

Views

Author

Emeric Deutsch, Aug 09 2006

Keywords

Examples

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

Links

Crossrefs

Cf. A100822.

Programs

  • Maple
    a[1]:=1: for n from 2 to 22 do a[n]:=a[n-1]+(n-1)!*(1+n*(n-1)/2) od: seq(a[n],n=1..22);

Formula

a(1) = 1, a(n) = a(n-1)+(n-1)!*(1+n*(n-1)/2) for n>=2.
a(n) = Sum_{k=1..n} k*A100822(n,k).
a(n) = (1/2)*Sum_{j=0..n+1} j! - n!. - Emeric Deutsch, Apr 06 2008
Conjecture D-finite with recurrence a(n) +(-n-4)*a(n-1) +3*(n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +2*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A121581 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=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.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 9, 11, 3, 1, 33, 43, 39, 4, 1, 153, 193, 199, 169, 5, 1, 873, 1057, 1099, 1081, 923, 6, 1, 5913, 6937, 7147, 7171, 7027, 6117, 7, 1, 46233, 53017, 54187, 54403, 54307, 53413, 47311, 8, 1, 409113, 461257, 468907, 470203, 470323, 469483, 463399
Offset: 1

Views

Author

Emeric Deutsch, Aug 11 2006

Keywords

Comments

Row sums are the factorials (A000142). T(n,0)=1; Sum(k*T(n,k), k=0..n)=A121582

Examples

			T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
Triangle starts:
1;
1,1;
1,3,2;
1,9,11,3;
1,33,43,39,4;

References

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

Crossrefs

Cf. A000142, A121582, A100822, A121583.

Programs

  • Maple
    Q[1]:=t: for n from 2 to 11 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))): P[1]:=1: P[n]:=subs(t=1,Q[n]): od: for n from 1 to 11 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form

Formula

The generating polynomial of row n is P(n,s)=Q(n,1,s), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.

Extensions

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

A121583 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the first two columns (n>=1, k>=1). 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

1, 0, 2, 0, 1, 4, 1, 0, 2, 6, 10, 5, 1, 0, 6, 16, 29, 34, 23, 11, 1, 0, 24, 60, 102, 148, 154, 119, 77, 35, 1, 0, 120, 288, 474, 668, 867, 874, 719, 533, 341, 155, 1, 0, 720, 1680, 2712, 3768, 4834, 5906, 5914, 5039, 4013, 2957, 1901, 875, 1, 0, 5040, 11520, 18360
Offset: 1

Views

Author

Emeric Deutsch, Aug 11 2006

Keywords

Comments

Row n has 2n-2 terms (n>=2). Row sums are the factorials (A000142). Sum(k*T(n,k), k=0..n)=A121584(n)

Examples

			T(2,2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having 2 cells in their first two columns.
Triangle starts:
1;
0,2;
0,1,4,1;
0,2,6,10,5,1;
0,6,16,29,34,23,11,1;

References

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

Crossrefs

Cf. A000142, A100822, A121581, A121584.

Programs

  • Maple
    Q[1]:=t: for n from 2 to 9 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))) od: for n from 1 to 9 do P[n]:=sort(subs(s=t,Q[n])): od: 1; for n from 1 to 9 do seq(coeff(P[n],t,j),j=1..2*n-2) od; # yields sequence in triangular form

Formula

The generating polynomial of row n is P(n,t)=Q(n,t,t), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.

A374574 a(n) = Sum_{j=n..2n} j!.

Original entry on oeis.org

1, 3, 32, 870, 46224, 4037880, 522956160, 93928267440, 22324392518400, 6780385526302080, 2561327494111411200, 1177652997443424902400, 647478071469567800985600, 419450149241406188889984000, 316196664211373618844934963200, 274410818470142134209609852672000
Offset: 0

Views

Author

Alois P. Heinz, Jul 11 2024

Keywords

Links

Crossrefs

Row sums of A143084.
Cf. A000142, A100822, A143122, A296591 (the same for product).
Diagonal of A054115, A211370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, [1, 3, 32][n+1],
     ((16*n^3-16*n^2-n+2)*a(n-1)-(n-1)*(16*n^3-20*n^2+6*n-1)
      *a(n-2)+2*(2*n-1)*(4*n+1)*(n-1)*(n-2)*a(n-3))/(4*n-3))
    end:
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) -(n-1)! +(2*n-1)! +(2*n)!)
    end:
seq(a(n), n=0..15);

Formula

a(n) = a(n-1) - (n-1)! + (2*n-1)! + (2*n)! with a(0) = 1.
a(n) = Sum_{j=0..n} (n + j)!.
a(n) = A100822(2n,n).
a(n) = A143122(2n,n).
Showing 1-4 of 4 results.

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