A331562 -id:A331562 - cp's OEIS frontend

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 43, 23, 1, 0, 0, 1, 374, 1879, 119, 1, 0, 0, 1, 3199, 173891, 102011, 719, 1, 0, 0, 1, 26945, 16140983, 117392909, 7235651, 5039, 1, 0, 0, 1, 224296, 1474050783, 142951955371, 117108036719, 674641325, 40319, 1

Offset: 0

Alois P. Heinz, Feb 19 2016

			Square array A(n,k) begins:
  0,   0,      0,         0,            0,               0, ...
  1,   0,      0,         0,            0,               0, ...
  1,   1,      1,         1,            1,               1, ...
  1,   5,     43,       374,         3199,           26945, ...
  1,  23,   1879,    173891,     16140983,      1474050783, ...
  1, 119, 102011, 117392909, 142951955371, 173996758190594, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Columns k=0-10 give: A057427, A033312, A267532, A269113, A269114, A269115, A269116, A269117, A269118, A269119, A269120.
Rows n=0-10 give: A000004, A000007, A000012, A269121, A269122, A269123, A269124, A269125, A269126, A269127, A269128.
Main diagonal gives: A268751.
Cf. A047909, A089759, A187783, A331562.

g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
      binomial(n-1, l[-1]-1)+add(f(sort(subsop(j=l[j]
      -1, l))), j=1..nops(l)-1))(add(i, i=l))
    end:
f:= l->(n->`if`(n=0, 1, `if`(l[1]=0, 0, `if`(n=1 or l[-1]=1, 1,
    `if`(n=2, binomial(l[1]+l[2], l[1])-1, g(l))))))(nops(l)):
A:= (n, k)-> (k*n)!/k!^n - f([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
      `if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
       j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
    end:
A:= (n, k)-> (n*k)!*(1/k!^n-b(n, k-1, 1, 0, irem(n, 2))*n!):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 03 2016

b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k == 0, (-1)^t/l!, If[p < 0, 0, Sum[b[k-i, p-1, j+1, l + i j, Mod[t + i j, 2]]/(i! p!^i), {i, 0, k}]] ];
A[n_, k_] := (n k)! (1/k!^n - b[n, k-1, 1, 0, Mod[n, 2]] n!); Table[ Table[ A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

A(n,k) = A089759(k,n) - A047909(k,n) = A187783(n,k) - A047909(k,n).

A103882 a(n) = Sum_{i=0..n} C(n+1,i)C(n-1,i-1)C(2n-i,n).

Original entry on oeis.org

1, 2, 12, 92, 780, 7002, 65226, 623576, 6077196, 60110030, 601585512, 6078578508, 61908797418, 634756203018, 6545498596110, 67830161708592, 705951252118284, 7375213677918294, 77310179609631564, 812839595630249540, 8569327862277434280, 90562666977432643862

Offset: 0

Ralf Stephan, Feb 20 2005

Number of permutations of n copies of 1..3 with all adjacent differences <= 1 in absolute value. - R. H. Hardin, May 06 2010 [Cf. A177316. - Peter Bala, Jan 14 2020]

Alois P. Heinz, Table of n, a(n) for n = 0..950 (terms n=1..94 from R. H. Hardin)
A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).

Cf. A005258, A156554, A177316, A352654.
Equals A103881(n, n).
Row n=3 of A331562.

[1] cat [&+[Binomial(n+1, i)*Binomial(n-1, i-1) * Binomial(2*n-i, n): i in [0..n]]:n in  [1..21]]; // Marius A. Burtea, Jan 19 2020

[&+[Binomial(n, k)^2*Binomial(n+k-1, k): k in [0..n]]:n in  [0..21]]; // Marius A. Burtea, Jan 19 2020

a:= proc(n) option remember; `if`(n<2, n+1,
      ((n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1)+
      n*(5*n-3)*(n-2)^2*a(n-2))/((n-1)*(5*n-8)*n^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 29 2015
# Alternative:
a := n -> hypergeom([-n, -n, n], [1, 1], 1):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Jan 19 2020

Drop[Table[Sum[Sum[Multinomial[r, g, n + 1 - r - g] Binomial[n - 1,n - r] Binomial[n - 1, n - g], {g, 1, n}], {r, 1, n}], {n, 0, 18}], 1] (* Geoffrey Critzer, Jun 29 2015 *)
Table[Sum[Binomial[n+1,k]Binomial[n-1,k-1]Binomial[2n-k,n],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jun 19 2021 *)

a(n) = polcoef(pollegendre(n, (1 + x)/(1 - x)) + O(x^(n+1)), n); \\ Michel Marcus, Dec 20 2020

def A103882(n):
    if n == 0: return 1
    m, g = 1, 0
    for k in range(n+1):
        g += m*n//(n+k)
        m *= (n+k+1)*(n-k)**2
        m //= (k+1)**3
    return g # Chai Wah Wu, Oct 04 2022

def A103882(n): return hypergeometric([-n,-n,n], [1,1], 1).simplify()
[A103882(n) for n in range(31)] # G. C. Greubel, May 24 2023

a(n) = (A005258(n-1) + 3*A005258(n))/5 (Apéry numbers). - Mark van Hoeij, Jul 13 2010
n^2*(n-1)*(5*n-8)*a(n) = (n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1) + n*(n-2)^2*(5*n-3)*a(n-2). - Alois P. Heinz, Jun 29 2015
a(n) ~ phi^(5*n + 3/2) / (2*Pi*5^(1/4)*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 21 2019
From Peter Bala, Jan 14 2020: (Start)
a(n) = Sum_{k = 0..n} C(n,k)^2*C(n+k-1,k). Cf. A005258.
For any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k (follows from known supercongruences satisfied by the Apéry numbers A005258 - see Straub, Example 3.4). (End)
a(n) = hypergeometric([-n, -n, n], [1, 1], 1). - Peter Luschny, Jan 19 2020
From Peter Bala, Dec 19 2020: (Start)
a(n) = Sum_{k = 1..n} C(n,k)*C(n+k,k)*C(n-1,k-1) for n >= 1.
a(n) = [x^n] P(n, (1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A156554. (End)
a(n) = Sum_{k = 0..n} binomial(2*n-k-1,n-k)*binomial(n,k)^2. Cf. A108628. - Peter Bala, Mar 24 2022
From Peter Bala, Apr 15 2022: (Start)
a(-n) = (-1)^n*A352654(n).
a(n) = [x^n*y^n*z^(n-1)] 1/(1 - x - y - z + x*z + y*z - x*y*z) for n >= 1.
a(n) = B(n,n,n-1) in the notation of Straub, see equation 24.
a(n) = [x^n*y^n*z^(n-1)] (x + y + z)^n*(x + y)^n*(y + z)^(n-1) for n >= 1. (End)
D-finite with recurrence 9*n^2*a(n) -3*(31*n^2-27*n+6)*a(n-1) -2*(37*n^2-138*n+108)*a(n-2) -(n-3)*(17*n-56)*a(n-3) -(n-4)^2*a(n-4) = 0. - R. J. Mathar, Aug 01 2022
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n-1, n-k)*binomial(n+k, k)*binomial(n+k-1, k). - Peter Bala, Aug 13 2023
a(n) = Sum_{k = 0..n} (-1)^k * binomial(n+1, k)*binomial(2*n-k, n-k)^2. - Peter Bala, Oct 05 2024

a(0)=1 prepended by Alois P. Heinz, Jun 29 2015

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.

Original entry on oeis.org

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

Alois P. Heinz, Aug 12 2016

			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, ...

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

Columns k=0-3 give: A000012, A000111, A275801, A276636.
Rows n=2-5 give: A000012, A241530, A036916, A276637.
Cf. A269129, A331562.

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);

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 *)

A177316 Number of permutations of n copies of 1..4 with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 2, 26, 506, 11482, 284002, 7426610, 201922730, 5650739930, 161686253810, 4708709084026, 139111173397066, 4159013698117618, 125595645802182818, 3825428523179727266, 117382025506323434506, 3625185567639373456090, 112597953571519245194770

Offset: 0

R. H. Hardin, May 06 2010

nonn

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 congruences 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

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

Cf. A005259, A103882, A177317 - A177328, A352653.

Row n=4 of A331562.

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
A177316 := n -> hypergeom([-n, -n, n, n], [1, 1, 1], 1):
seq(simplify(A177316(n)), n = 0..17); # Peter Luschny, Mar 27 2023

a[n_] := HypergeometricPFQ[{-n, -n, n, n}, {1, 1, 1}, 1];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, May 28 2023, after Peter Luschny *)

def A177316(n):
    if n == 0: return 1
    m, g = 1, 0
    for k in range(n+1):
        g += m*n**2//(n+k)**2
        m *= ((n+k+1)*(n-k))**2
        m //= (k+1)**4
    return g # Chai Wah Wu, Oct 03 2022

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. [added Apr 18 2022: assuming the recurrence given in the Maple program below is correct then these conjectures are true.] (End)
a(n) = 2*A352653(n) for n >= 1. - Peter Bala, Apr 18 2022
a(n) = hypergeom([-n, -n, n, n], [1, 1, 1], 1). - Peter Luschny, Mar 27 2023
a(n) ~ (1 + sqrt(2))^(4*n) / (2^(5/4) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 29 2023

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

A208673 Number of words A(n,k), either empty or beginning with the first letter of the k-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 10, 1, 1, 1, 1, 9, 37, 35, 1, 1, 1, 1, 15, 163, 309, 126, 1, 1, 1, 1, 25, 640, 3593, 2751, 462, 1, 1, 1, 1, 41, 2503, 36095, 87501, 25493, 1716, 1, 1, 1, 1, 67, 9559, 362617, 2336376, 2266155, 242845, 6435, 1, 1

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the absolute difference of the dimension indices used in consecutive steps is <= 1.

All rows are linear recurrences with constant coefficients and for n > 0 the order of the recurrence is bounded by 2*n-1. For n up to at least 20 this upper bound is exact. - Andrew Howroyd, Feb 22 2022

			A(0,0) = A(n,0) = A(0,k) = 1: the empty word.
A(2,3) = 5:
  +------+   +------+   +------+   +------+   +------+
  |aabbcc|   |aabcbc|   |aabccb|   |ababcc|   |abccba|
  +------+   +------+   +------+   +------+   +------+
  |122222|   |122222|   |122222|   |112222|   |111112|
  |001222|   |001122|   |001112|   |011222|   |011122|
  |000012|   |000112|   |000122|   |000012|   |001222|
  +------+   +------+   +------+   +------+   +------+
  |xx    |   |xx    |   |xx    |   |x x   |   |x    x|
  |  xx  |   |  x x |   |  x  x|   | x x  |   | x  x |
  |    xx|   |   x x|   |   xx |   |    xx|   |  xx  |
  +------+   +------+   +------+   +------+   +------+
Square array A(n,k) begins:
  1,  1,    1,     1,       1,         1,           1, ..
  1,  1,    1,     1,       1,         1,           1, ..
  1,  1,    3,     5,       9,        15,          25, ..
  1,  1,   10,    37,     163,       640,        2503, ..
  1,  1,   35,   309,    3593,     36095,      362617, ..
  1,  1,  126,  2751,   87501,   2336376,    62748001, ..
  1,  1,  462, 25493, 2266155, 164478048, 12085125703, ..

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns k=0+1, 2-4 give: A000012, A088218, A208675, A212334.
Rows n=0+1, 2-3 give: A000012, A001595, A208674.
Main diagonal gives A351759.
Cf. A208879 (cyclic alphabet), A331562.

b:= proc(t, l) option remember; local n; n:= nops(l);
     `if`(n<2 or {0}={l[]}, 1,
     `if`(l[t]>0, b(t, [seq(l[i]-`if`(i=t, 1, 0), i=1..n)]), 0)+
     `if`(t0,
                  b(t+1, [seq(l[i]-`if`(i=t+1, 1, 0), i=1..n)]), 0)+
     `if`(t>1 and l[t-1]>0,
                  b(t-1, [seq(l[i]-`if`(i=t-1, 1, 0), i=1..n)]), 0))
    end:
A:= (n, k)-> `if`(n=0 or k=0, 1, b(1, [n-1, n$(k-1)])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[t_, l_List] := b[t, l] = Module[{n = Length[l]}, If[n < 2 || {0} == Union[l], 1, If[l[[t]] > 0, b[t, Table[l[[i]] - If[i == t, 1, 0], {i, 1, n}]], 0] + If[t < n && l[[t + 1]] > 0, b[t + 1, Table[l[[i]] - If[i == t + 1, 1, 0], {i, 1, n}]], 0] + If[t > 1 && l[[t - 1]] > 0, b[t - 1, Table[l[[i]] - If[i == t - 1, 1, 0], {i, 1, n}]], 0]]]; A[n_, k_] := If[n == 0 || k == 0, 1, b[1, Join[{n - 1}, Array[n&, k - 1]]]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

F(u)={my(n=#u); sum(k=1, n,u[k]*binomial(n-1,k-1))}
step(u, c)={my(n=#u); vector(n, k, sum(i=max(0, 2*k-c-n), k-1, sum(j=0, n-2*k+i+c, u[k-i+j]*binomial(n-1, 2*k-1-c-i+j)*binomial(k-1, k-i-1)*binomial(k-i+j-c, j) ))) }
R(n,k)={my(r=vector(n+1), u=vector(k), v=vector(k)); u[1]=v[1]=r[1]=r[2]=1; for(n=3, #r, u=step(u,1); v=step(v,0)+u; r[n]=F(v)); r}
T(n,k)={if(n==0||k==0, 1, R(k,n)[1+k])} \\ Andrew Howroyd, Feb 22 2022

A177282 Number of permutations of 2 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 6, 12, 26, 48, 86, 148, 250, 416, 686, 1124, 1834, 2984, 4846, 7860, 12738, 20632, 33406, 54076, 87522, 141640, 229206, 370892, 600146, 971088, 1571286, 2542428, 4113770, 6656256, 10770086, 17426404, 28196554, 45623024, 73819646, 119442740, 193262458

Offset: 0

R. H. Hardin, May 06 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..197 from R. H. Hardin)

Column k=2 of A331562.

Cf. A000680.

a:= proc(n) option remember; `if`(n<4, [1$2, 6, 12][n+1],
     ((8*n-31)*a(n-1) -(4*n-19)*a(n-2) -(3*n-10)*a(n-3)
      +(2*n-10)*a(n-4)) / (3*n-11))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 14 2016

a(n) = (2n)!/2^n = A000680(n) for n<=2.

A177317 Number of permutations of n copies of 1..5 with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 2, 48, 2288, 135040, 8956752, 640160976, 48203722464, 3772321496064, 304100156874800, 25098440923318048, 2111488538062121088, 180477438192133215952, 15633823902235680250592, 1369837117884520736235840, 121216041295339359662340288, 10819157637786569144853012480

Offset: 0

R. H. Hardin, May 06 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..28 (terms n=1..16 from R. H. Hardin)
Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 9-11.

Row n=5 of A331562.

From Manuel Kauers and Christoph Koutschan, Mar 01 2023: (Start)
a(n) = coefficient of x1^n*x2^n*x3^n*x4^n*t^(5*n-1) in (2*t^3*x3(x1*x2+x1*x4*x2+x4*x2+x1*x4)-t^2*x3*(x2*x1-3*x1+x2*x4+x4)-2*t*(x3*x1+x4*x1+x1+x2+x3+x2*x4)+x1+x2+x3+x4+1)/(-t^4*x3*(x1*x2+x1*x4*x2+x4*x2+x1*x4)+t^3*x3*(x2*x1-x1+x2*x4+x4)+t^2*(x3*x1+x4*x1+x1+x2+x3+x2*x4)-t*(x1+x2+x3+x4+1)+1).
3*n^3*(1 + n)*(1 + 3*n)*(2 + 3*n)*(3281160 + 13324928*n + 23607946*n^2 + 23825758*n^3 + 14975281*n^4 + 6000286*n^5 + 1496236*n^6 + 212252*n^7 + 13113*n^8)*a(n) - (1 + n)^2*(14722560 + 163505952*n + 822949992*n^2 + 2464399296*n^3 + 4847819730*n^4 + 6543447222*n^5 + 6186525969*n^6 + 4125650658*n^7 + 1929434771*n^8 + 618883678*n^9 + 129652375*n^10 + 15978026*n^11 + 878571*n^12)*a(n+1) + 2*(2 + n)^2*(20370096 + 207973548*n + 951883014*n^2 + 2588508450*n^3 + 4659341433*n^4 + 5838584798*n^5 + 5211702571*n^6 + 3333874350*n^7 + 1515722000*n^8 + 477646252*n^9 + 99089547*n^10 + 12162378*n^11 + 668763*n^12)*a(n+2) - (2 + n)^2*(3 + n)^4*(10512 + 90060*n + 332910*n^2 + 697266*n^3 + 906481*n^4 + 745834*n^5 + 377636*n^6 + 107348*n^7 + 13113*n^8)*a(n+3) = 0. (End)

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

A177291 Number of permutations of 3 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 20, 92, 506, 2288, 10010, 41618, 168284, 664958, 2584442, 9916688, 37679618, 142079906, 532572428, 1987037318, 7386724082, 27381500624, 101272019258, 373902595130, 1378571667644, 5077289249390, 18683930010890, 68709775705328, 252549056389394, 927895845621746

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (3n)!/6^n = A014606(n) for n<=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=3 of A331562.

Cf. A014606.

a(0)=1 prepended by Alois P. Heinz, Jan 21 2020
a(19)-a(20) from Alois P. Heinz, Jan 22 2020
Terms a(21) and beyond from Andrew Howroyd, May 14 2020

A177298 Number of permutations of 4 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 70, 780, 11482, 135040, 1543862, 16699380, 175280570, 1792874048, 17990831182, 177773616180, 1735220375914, 16769137076968, 160740670148270, 1530462874512756, 14491157235576770, 136575590461035368, 1282229192334549022, 11999276440029403356, 111987255233226452770

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (4n)!/24^n = A014608(n) for n<=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A014608.

Column k=4 of A331562.

a(0)=1 prepended by Alois P. Heinz, Jan 20 2020
a(12)-a(13) from Alois P. Heinz, Jan 21 2020
Terms a(14) and beyond from Andrew Howroyd, May 15 2020

A177301 Number of permutations of 5 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 252, 7002, 284002, 8956752, 276285002, 8039989002, 226901044252, 6232521421502, 167765555015002, 4441865811412752, 116042171156004002, 2998175631045512002, 76751828260736962252, 1949598912568702682502, 49197330547669842130002, 1234523426440511782592752

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (5n)!/120^n = A014609(n) for n<=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A014609.

Column k=5 of A331562.

a(10) from Alois P. Heinz, Jan 21 2020

Terms a(11) and beyond from Andrew Howroyd, May 15 2020

cp's OEIS Frontend

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A103882 a(n) = Sum_{i=0..n} C(n+1,i)*C(n-1,i-1)*C(2n-i,n).

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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.

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A177316 Number of permutations of n copies of 1..4 with all adjacent differences <= 1 in absolute value.

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A208673 Number of words A(n,k), either empty or beginning with the first letter of the k-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A177282 Number of permutations of 2 copies of 1..n with all adjacent differences <= 1 in absolute value.

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A177317 Number of permutations of n copies of 1..5 with all adjacent differences <= 1 in absolute value.

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A103882 a(n) = Sum_{i=0..n} C(n+1,i)C(n-1,i-1)C(2n-i,n).