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A177316 Number of permutations of n copies of 1..4 with all adjacent differences <= 1 in absolute value. 3
1, 2, 26, 506, 11482, 284002, 7426610, 201922730, 5650739930, 161686253810, 4708709084026, 139111173397066, 4159013698117618, 125595645802182818, 3825428523179727266, 117382025506323434506, 3625185567639373456090, 112597953571519245194770 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See A103882 and A177317 through A177328 for the number of permutations of n copies of 1..k (for different values of k) with adjacent differences restricted in size. We conjecture that all these sequences satisfy the supercongruences A(n*p^k) == A(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k and any prime p >= 5. - Peter Bala, Jan 16 2020

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..656 (terms n=1..31 from R. H. Hardin)

FORMULA

From Peter Bala, Jan 14 2020: (Start)

Conjecture: a(n) = (1/3)*( A005259(n) + A005259(n-1) ).

Equivalently, a(n) = Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k-1,k)^2. Cf. A103882. If true, then the sequence satisfies the recurrence a(n) = (2*(102*n^6 - 612*n^5 + 1462*n^4 - 1768*n^3 + 1143*n^2 - 382*n+52) * a(n-1) - (2*n-1)*(3*n^2 - 3*n+1) * (n-2)^3 * a(n-2)) / (n^3*(2*n - 3) * (3*n^2 - 9*n+7)) and the supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k and any prime p >= 5. (End)

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 2, 26][n+1],

       (3*((105*n^4-356*n^3+402*n^2-208*n+43)*a(n-1)

      -(105*n^4-904*n^3+2868*n^2-3932*n+1930)*a(n-2))

      +(9*n-11)*(n-3)^3*a(n-3))/((9*n-16)*n^3))

    end:

seq(a(n), n=0..23);  # Alois P. Heinz, Jan 22 2020

CROSSREFS

Cf. A005259, A103882, A177317 - A177328.

Row n=4 of A331562.

Sequence in context: A137100 A228411 A216254 * A255538 A302719 A090247

Adjacent sequences:  A177313 A177314 A177315 * A177317 A177318 A177319

KEYWORD

nonn

AUTHOR

R. H. Hardin, May 06 2010

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jan 20 2020

STATUS

approved

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Last modified June 16 04:56 EDT 2021. Contains 345056 sequences. (Running on oeis4.)