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 1-2 of 2 results.

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.

A304680 Total number of tilings of Ferrers-Young diagrams using dominoes and at most one monomino summed over all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 6, 23, 16, 76, 42, 239, 106, 688, 268, 1931, 650, 5266, 1580, 13861, 3750, 35810, 8862, 91065, 20598, 226914, 47776, 559271, 109248, 1360152, 248966, 3270429, 562630, 7785974, 1264780, 18378067, 2823958, 43007532, 6282198, 99892837, 13884820
Offset: 0

Views

Author

Alois P. Heinz, May 16 2018

Keywords

Links

Crossrefs

Bisection (even part) gives A304662.
Cf. A304677.

Programs

  • Maple
    h:= proc(l, f, t) 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), t))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(t, h(subsop(k=1, l), f, false), 0)+
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f, t), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f, t), 0)
      fi
    end:
g:= l-> (t-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l),
               is(t, odd))))(add(i, i=l)):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> b(n$2, []):
seq(a(n), n=0..23);
Showing 1-2 of 2 results.

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