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.

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

Offset: 0

Author

Alois P. Heinz, Jan 20 2020

Keywords

• nonn
• tabl

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

Row 2, the sequence of central binomial numbers A000984, satisfies the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r (see Meštrović, equation 39). This is also known to be true for row 3 (A103882) and row 4 (A177316). We conjecture that each row sequence of the table satisfies the same congruences. - Peter Bala, Oct 26 2024.

Examples

			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, ...
		

Links

• Andrew Howroyd, Antidiagonals n = 0..50, flattened (antidiagonals 0..15 from Alois P. Heinz)
• Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Crossrefs

Columns k=0-9 give: A000012, A130130 (for n>0), A177282, A177291, A177298, A177301, A177304, A177307, A177310, A177313.

Rows n=0+1, 2-9 give: A000012, A000984, A103882, A177316, A177317, A177318, A177319, A177320, A177321.

Main diagonal gives A331623.

Cf. A269129, A275784.

Programs

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 *)

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