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A331562 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding absolute differences between adjacent elements larger than one; square array A(n,k), n>=0, k>=0, read by antidiagonals. 20
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 1, 20, 12, 2, 1, 1, 1, 70, 92, 26, 2, 1, 1, 1, 252, 780, 506, 48, 2, 1, 1, 1, 924, 7002, 11482, 2288, 86, 2, 1, 1, 1, 3432, 65226, 284002, 135040, 10010, 148, 2, 1, 1, 1, 12870, 623576, 7426610, 8956752, 1543862, 41618, 250, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
All columns are linear recurrences with constant coefficients and for k > 0 the order of the recurrence is bounded by 3*k-1. For k up to at least 17 this upper bound is exact. - Andrew Howroyd, May 16 2020
LINKS
Andrew Howroyd, Antidiagonals n = 0..50, flattened (antidiagonals 0..15 from Alois P. Heinz)
EXAMPLE
A(2,2) = 6: 1122, 1212, 1221, 2112, 2121, 2211.
A(3,2) = 12: 112233, 112323, 112332, 121233, 123321, 211233, 233211, 321123, 323211, 332112, 332121, 332211.
A(2,3) = 20: 111222, 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,3) = 92: 111222333, 111223233, 111223323, 111223332, ..., 333221112, 333221121, 333221211, 333222111.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 6, 20, 70, 252, 924, ...
1, 2, 12, 92, 780, 7002, 65226, ...
1, 2, 26, 506, 11482, 284002, 7426610, ...
1, 2, 48, 2288, 135040, 8956752, 640160976, ...
1, 2, 86, 10010, 1543862, 276285002, 54331653686, ...
MAPLE
b:= proc(l, q) option remember; (n-> `if`(n<2, 1, add(
`if`(l[j]=1, `if`(j in [1, n], b(subsop(j=[][], l),
`if`(j=1, 0, n)), 0), b(subsop(j=l[j]-1, l), j)), j=
`if`(q<0, 1..n, max(1, q-1)..min(n, q+1)))))(nops(l))
end:
A:= (n, k)-> `if`(k=0, 1, b([k$n], -1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[l_, q_] := b[l, q] = With[{n = Length[l]}, If[n < 2, 1, Sum[
If[l[[j]] == 1, If[j == 1 || j == n, b[ReplacePart[l, j -> Nothing],
If[j == 1, 0, n]], 0], b[ReplacePart[l, j -> l[[j]] - 1], j]], {j,
If[q < 0, Range[n], Range[Max[1, q - 1], Min[n, q + 1]]]}]]];
A[n_, k_] := If[k == 0, 1, b[Table[k, {n}], -1]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)
PROG
(PARI)
step(m, R)={my(M=matrix(3, m+1, q, p, q--; p--; sum(j=0, m-p-q, sum(i=max(p+j-#R+1, 2*p+q+j-m), p, R[1+q, 1+p+j-i] * binomial(p, i) * binomial(p+q+j-i-1, j) * binomial(m-1, 2*p+q+j-i-1))))); M[3, ]+=2*M[2, ]+M[1, ]; M[2, ]+=M[1, ]; M}
AdjPathsBySig(sig)={if(#sig<1, 1, my(R=[1; 1; 1]); for(i=1, #sig-1, R=step(sig[i], R)); my(m=sig[#sig]); sum(i=1, min(m, #R), binomial(m-1, i-1)*R[3, i]))}
T(n, k) = {if(k==0, 1, AdjPathsBySig(vector(n, i, k)))} \\ Andrew Howroyd, May 16 2020
CROSSREFS
Columns k=0-9 give: A000012, A130130 (for n>0), A177282, A177291, A177298, A177301, A177304, A177307, A177310, A177313.
Main diagonal gives A331623.
Sequence in context: A060185 A348091 A129110 * A257101 A112624 A294875
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 20 2020
STATUS
approved

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Last modified April 17 08:54 EDT 2024. Contains 371763 sequences. (Running on oeis4.)