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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.

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A304662 Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.

Original entry on oeis.org

1, 2, 6, 16, 42, 106, 268, 650, 1580, 3750, 8862, 20598, 47776, 109248, 248966, 562630, 1264780, 2823958, 6282198, 13884820, 30590124, 67051982, 146463790, 318588916, 690882926, 1492592450, 3215372064, 6904561416, 14786529836, 31574656096, 67261524262
Offset: 0

Views

Author

Alois P. Heinz, May 16 2018

Keywords

Examples

			a(2) = 6:
._.  .___.  ._._.  .___.  ._.___.  .___.___.
| |  |___|  | | |  |___|  | |___|  |___|___|
|_|  | |    |_|_|  |___|  |_|
| |  |_|
|_|

Links

Crossrefs

Row sums of A304718.
Bisection (even part) of A304680.
Cf. A004003, A296625, A300060, A300789, A304677, A304710, A304790.

Programs

  • Maple
    h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> b(2*n$2, []):
seq(a(n), n=0..12);
  • Mathematica
    h[l_, f_] := h[l, f] = Module[{k}, If[Min[l]>0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]]]]-1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]]>0, k--]; If[Length[f]>0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k>1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k-1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i-1, l] + b[n-i, Min[n-i, i], Append[l, i]]];
a[n_] := b[2n, 2n, {}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..A304790(n)} k * A304789(n,k).
a(n) = Sum_{k=0..n} A304718(n,k).
a(n) = A296625(n) for n < 7.

A304719 Number of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly floor(n/2) horizontally oriented dominoes.

Original entry on oeis.org

1, 1, 2, 5, 14, 28, 62, 150, 380, 787, 1760, 3951, 9338, 19536, 43224, 94326, 213278, 448193, 979712, 2094981, 4622262, 9670378, 20886560, 44067191, 95469402, 198712506
Offset: 0

Views

Author

Alois P. Heinz, May 17 2018

Keywords

Examples

			a(3) = 5:
:  .___.  ._.___.  .___.  ._._.  ._._.___.
:  |___|  | |___|  |___|  | | |  | | |___|
:  | |    |_|      | | |  |_|_|  |_|_|
:  |_|    | |      |_|_|  |___|
:  | |    |_|
:  |_|

Links

Crossrefs

Cf. A304718.

Programs

  • Maple
    h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od; expand(
        `if`(nops(f)>0 and f[1]>=k, x*h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0))
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> coeff(b(2*n$2, []), x, iquo(n, 2)):
seq(a(n), n=0..14);
  • Mathematica
    h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]]]] - 1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]]>0, k--]; If[Length[f] > 0 && f[[1]] >= k, x*h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k - 1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i-1, l] + b[n-i, Min[n-i, i], Append[l, i]]];
T[n_] := CoefficientList[b[2 n, 2 n, {}], x];
a[n_] := T[n][[Floor[n/2] + 1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A304718 *)

Formula

a(n) = A304718(n,floor(n/2)).
Showing 1-2 of 2 results.

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