A275784 Number A(n,k) of up-down sequences with k copies each of 1,2,...,n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 5, 1, 1, 0, 1, 12, 53, 16, 1, 1, 0, 1, 36, 761, 936, 61, 1, 1, 0, 1, 120, 12661, 87336, 25325, 272, 1, 1, 0, 1, 400, 229705, 9929000, 18528505, 933980, 1385, 1, 1, 0, 1, 1400, 4410665, 1267945800, 17504311533, 6376563600, 45504649, 7936, 1

Offset

0,14

Links

Alois P. Heinz, Antidiagonals n = 0..15, flattened

Example

A(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
A(3,2) = 4: 121323, 132312, 231213, 231312.
A(3,3) = 12: 121313232, 121323132, 121323231, 131213232, 132312132, 132323121, 231213132, 231213231, 231312132, 231323121, 232312131, 232313121.
A(2,4) = 1: 12121212.
Square array A(n,k) begins:
  1,   1,      1,          1,              1,              1, ...
  1,   1,      0,          0,              0,              0, ...
  1,   1,      1,          1,              1,              1, ...
  1,   2,      4,         12,             36,            120, ...
  1,   5,     53,        761,          12661,         229705, ...
  1,  16,    936,      87336,        9929000,     1267945800, ...
  1,  61,  25325,   18528505,    17504311533, 19126165462061, ...
  1, 272, 933980, 6376563600, 59163289699260, ...

Maple

b:= proc(n, l) option remember; `if`(l=[], 1, `if`(irem(add(i,
      i=l), 2)=0, add(b(i, subsop(i=`if`(l[i]=1, [][], l[i]-1),
      l)), i=n+1..nops(l)), add(b(i-`if`(l[i]=1, 1, 0), subsop(
      i=`if`(l[i]=1, [][], l[i]-1), l)), i=1..n-1)))
    end:
A:= (n, k)->`if`(k=0, 1, b(`if`(irem(k*n, 2)=0, 0, n+1), [k$n])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

b[n_, l_List] := b[n, l] = If[l == {}, 1, If[EvenQ[Total[l]], Sum[b[i, ReplacePart[l, i -> If[l[[i]] == 1, Nothing, l[[i]]-1]]], {i, n+1, Length[l]}], Sum[b[i - If[l[[i]] == 1, 1, 0], ReplacePart[l, i -> If[l[[i]] == 1, Nothing, l[[i]]-1]]], {i, 1, n-1}]]]; A[n_, k_] := If[k == 0, 1, b[If[EvenQ[k*n], 0, n+1], Array[k&, n]]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 23 2017, adapted from Maple *)

Crossrefs

Columns k=0-3 give: A000012, A000111, A275801, A276636.

Rows n=2-5 give: A000012, A241530, A036916, A276637.

Cf. A269129, A331562.

Sequence in context: A108947 A338859 A152459 * A331508 A097608 A331126

Adjacent sequences: A275781 A275782 A275783 * A275785 A275786 A275787

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 12 2016

Status

approved