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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..15, flattened
Crossrefs
Programs
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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 *)