A208673 -id:A208673 - cp's OEIS frontend

A212334 Number of words, either empty or beginning with the first letter of the 4-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

1, 1, 9, 163, 3593, 87501, 2266155, 61211095, 1704838665, 48605519665, 1411522695509, 41606511550803, 1241591466423467, 37435593955828069, 1138713916992923679, 34901292375152457663, 1076813644170756916745, 33416749492077957930105, 1042376218505671236116985

Offset: 0

Alois P. Heinz, Aug 07 2012

nonn

Also the number of (4*n-1)-step walks on 4-dimensional cubic lattice from (1,0,0,0) to (n,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.
It appears that for primes p >= 5, a(p) == 1 (mod p^5). Cf. A352655. - Peter Bala, Dec 12 2021
Conjecture: for r >= 2, and all primes p >= 5, a(p^r) == a(p^(r-1)) (mod p^(3*r+3)). - Peter Bala, Oct 13 2022

Alois P. Heinz, Table of n, a(n) for n = 0..656

Column k = 4 of A208673.

Cf. A005259, A352655.

a:= proc(n) option remember; `if`(n<3, [1, 1, 9][n+1],
      ((26682*n^4 -102687*n^3 +149385*n^2 -109413*n +31101) *a(n-1)
      +(-161058*n^4 +1392915*n^3 -4418826*n^2 +6030348*n -2931516) *a(n-2)
      +(4718*n^4 -47957*n^3 +176841*n^2 -275751*n +148365) *a(n-3)) /
      (n^3 *(646*n -1057)))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n < 3, {1, 1, 9}[[n + 1]], ((26682 n^4 - 102687 n^3 + 149385 n^2 - 109413 n + 31101) a[n-1] + (-161058 n^4 + 1392915 n^3 - 4418826 n^2 + 6030348 n - 2931516)a[n-2] + (4718 n^4 - 47957 n^3 + 176841 n^2 - 275751 n + 148365)a[n-3])/(n^3 (646 n - 1057))];
a /@ Range[0, 30] (* Jean-François Alcover, May 14 2020, after Maple *)

a(n) ~ (1 + sqrt(2))^(4*n-1) / (2^(7/4) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 13 2013, simplified Apr 06 2022
From Peter Bala, Apr 17 2022: (Start)
a(n) = (1/12)*(A005259(n) + 7*A005259(n-1)) for n >= 1.
The supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
a(n) = (1/3)*Sum_{k = 0..n} binomial(n,k)^2*binomial(n + k,k)^2*(2*n^2 - 3*k*n + 2*k^2)/(n + k)^2.
(24*n^3 - 102*n^2 + 148*n - 73)*n^3*a(n) = 4*(204*n^6 - 1173*n^5 + 2668*n^4 - 3065*n^3 + 1905*n^2 - 634*n + 86)*a(n-1) - (24*n^3 - 30*n^2 + 16*n-3)*(n - 2)^3*a(n-2) with a(0) = a(1) = 1. (End)
a(n) = Sum_{k=0..n-1} binomial(n,k)*binomial(n-1,k)*binomial(n+k-1,k)^2 for n>=1. - Peter Bala, Mar 22 2023

A208675 Number of words, either empty or beginning with the first letter of the ternary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

Original entry on oeis.org

1, 1, 5, 37, 309, 2751, 25493, 242845, 2360501, 23301307, 232834755, 2349638259, 23905438725, 244889453043, 2523373849701, 26132595017037, 271826326839477, 2838429951771795, 29740725671232119, 312573076392760183, 3294144659048391059, 34802392680979707121

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (3*n-1)-step walks on 3-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.

			a(2) = 5 = |{aabbcc, aabcbc, aabccb, ababcc, abccba}|.
a(3) = 37 = |{aaabbbccc, aaabbcbcc, aaabbccbc, aaabbcccb, aaabcbbcc, aaabcbcbc, aaabcbccb, aaabccbbc, aaabccbcb, aaabcccbb, aababbccc, aababcbcc, aababccbc, aababcccb, aabbabccc, aabbcccba, aabcbabcc, aabcbccba, aabccbabc, aabccbcba, aabcccbab, aabcccbba, abaabbccc, abaabcbcc, abaabccbc, abaabcccb, abababccc, ababcccba, abbaabccc, abbcccbaa, abcbaabcc, abcbccbaa, abccbaabc, abccbcbaa, abcccbaab, abcccbaba, abcccbbaa}|.

Alois P. Heinz, Table of n, a(n) for n = 0..960
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.

Column k=3 of A208673.

Cf. A000108, A001622, A005258, A108625, A108628, A271777.

A208675:= func< n | (&+[Binomial(n,j)*Binomial(n-1,j)*Binomial(n+j-1,j): j in [0..2*n]]) >;
[A208675(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

a:= n-> add(binomial(n-1, k)^2 *binomial(2*n-1-k, n-k), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 26 2012

a[n_]:= HypergeometricPFQ[{1-n,-n,n}, {1,1}, 1] (* Michael Somos, Jun 03 2012 *)

def A208675(n): return sum(binomial(n,j)*binomial(n-1,j)*binomial(n+j-1,j) for j in range(n+1))
[A208675(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

From Michael Somos, Jun 03 2012: (Start)
a(n) = A108625(n-1, n).
a(n) = Hypergeometric3F2([1-n, -n, n], [1, 1], 1).
(n+1)^2 * (1 -4*n +5*n^2) * a(n+1) = (5 -5*n -26*n^2 +11*n^3 +55*n^4) * a(n) + (n-1)^2 * (2 +6*n +5*n^2) * a(n-1). (End)
a(n) ~ sqrt((5-sqrt(5))/10)/(2*Pi*n) * ((1+sqrt(5))/2)^(5*n). - Vaclav Kotesovec, Dec 06 2012. Equivalently, a(n) ~ phi^(5*n - 1/2) / (2 * 5^(1/4) * Pi * n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 15*x^3 + 94*x^4 + 668*x^5 + 5144*x^6 + 41884*x^7 + 355307*x^8 + ... appears to have integer coefficients. Cf. A108628. - Peter Bala, Jan 12 2016
From Peter Bala, Apr 05 2022: (Start)
a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(n-1,k)*binomial(n+k-1,k).
Using binomial(-n,k) = (-1)^k*binomial(n+k-1,k) for nonnegative k, we have:
a(-n) = Sum_{k = 0..n} binomial(-n,k)*binomial(-n-1,k)*binomial(-n+k-1,k).
a(-n) = Sum_{k = 0..n} (-1)^k* binomial(n+k-1,k)*binomial(n+k,k)*binomial(n,k)
a(-n) = (-1)^n*A108628(n-1), for n >= 1.
a(n) = Sum_{k = 1..n} binomial(n,k)*binomial(n-1,k-1)*binomial(n+k-1,k-1) for n >= 1.
Equivalently, a(n) = [(x^n)*(y*z)^(n-1)] (x + y + z)^n*(x + y)^(n-1)*(y + z)^(n-1) for n >= 1.
a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n-1,k)*binomial(2*n-k-1,n-k)^2.
a(n) = (1/5)*(A005258(n) + 2*A005258(n-1)) for n >= 1.
a(n) = [x^n] 1/(1 - x)*P(n-1,(1 + x)/(1 - x)) for n >= 1, where P(n,x) denotes the n-th Legendre polynomial. Compare with A005258(n) = [x^n] 1/(1 - x)*P(n,(1 + x)/(1 - x)).
a(n) = B(n,n-1,n-1) in the notation of Straub, equation 24. Hence
a(n) = [(x^n)*(y*z)^(n-1)] 1/(1 - x - y - z + x*z + y*z - x*y*z) for n >= 1.
The supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
Conjectures:
1) a(n) = [(x*y)^n*z^(n-1)] 1/(1 - x - y - z + x*y + x*y*z) for n >= 1.
2) a(n) = - [(x*z)^(n-1)*(y^n)] 1/(1 + y + z + x*y + y*z + x*z + x*y*z) for n >= 1.
3) a(n) = [x^(n-1)*(y*z)^n] 1/(1 - x - x*y - y*z - x*z - x*y*z) for n >= 1. (End)
From Peter Bala, Mar 17 2023: (Start)
For n >= 1:
a(n) = Sum_{k = 0..n} ((n-k)/(n+k))*binomial(n,k)^2*binomial(n+k,k).
a(n) = Sum_{k = 0..n} (-1)^(n+k-1) * ((n-k)/(n+k)) * binomial(n,k) * binomial(n+k,k)^2. (End)

A208879 Number of words A(n,k), either empty or beginning with the first letter of the cyclic 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, 2, 3, 1, 1, 1, 2, 30, 10, 1, 1, 1, 2, 62, 560, 35, 1, 1, 1, 2, 114, 2830, 11550, 126, 1, 1, 1, 2, 202, 12622, 151686, 252252, 462, 1, 1, 1, 2, 346, 53768, 1754954, 8893482, 5717712, 1716, 1, 1

Offset: 0

Alois P. Heinz, Mar 02 2012

The first and the last letters are considered neighbors in a cyclic alphabet. The words are not considered cyclic here.

A(n,k) is 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 indices of dimensions used in consecutive steps differ by less than 2 or are in the set {1,k}.

			A(0,0) = A(n,0) = A(0,k) = 1: the empty word.
A(1,2) = 1 = |{ab}|.
A(1,3) = 2 = |{abc, acb}|.
A(1,4) = 2 = |{abcd, adcb}|.
A(2,2) = 3 = |{aabb, abab, abba}|.
A(2,4) = 62 = |{aabbccdd, aabbcdcd, aabbcddc, aabcbcdd, aabcddcb, aadcbbcd, aadcdcbb, aaddcbbc, aaddcbcb, aaddccbb, ababccdd, ababcdcd, ababcddc, abadcbcd, abadcdcb, abaddcbc, abaddccb, abbadccd, abbadcdc, abbaddcc, abbccdad, abbccdda, abbcdadc, abbcdcda, abcbadcd, abcbaddc, abcbcdad, abcbcdda, abccbadd, abccddab, abcdabcd, abcdadcb, abcdcbad, abcdcdab, abcddabc, abcddcba, adabbccd, adabbcdc, adabcbcd, adabcdcb, adadcbbc, adadcbcb, adadccbb, adcbabcd, adcbadcb, adcbbadc, adcbbcda, adcbcbad, adcbcdab, adccbbad, adccdabb, adcdabbc, adcdabcb, adcdcbab, adcdcbba, addabbcc, addabcbc, addabccb, addcbabc, addcbcba, addccbab, addccbba}|.
Square array A(n,k) begins:
  1,  1,   1,       1,         1,           1,             1, ...
  1,  1,   1,       2,         2,           2,             2, ...
  1,  1,   3,      30,        62,         114,           202, ...
  1,  1,  10,     560,      2830,       12622,         53768, ...
  1,  1,  35,   11550,    151686,     1754954,      19341130, ...
  1,  1, 126,  252252,   8893482,   276049002,    8151741752, ...
  1,  1, 462, 5717712, 552309938, 46957069166, 3795394507240, ...

Columns k=0+1, 2-6 give: A000012, A088218, A208881, A209183, A209184, A209185.
Rows n=0, 2 give: A000012, A208880.
Cf. A208673 (noncyclic alphabet).

b:= proc() option remember; local n; n:= nargs;
     `if`(n<2 or {0}={args}, 1,
     `if`(n=2, `if`(args[1]>0, b(args[1]-1, args[2]), 0)+
               `if`(args[2]>0, b(args[2]-1, args[1]), 0),
     `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..n)), 0)+
     `if`(args[2]>0, b(args[2]-1, seq(args[i], i=3..n), args[1]), 0)+
     `if`(args[n]>0, b(args[n]-1, seq(args[i], i=1..n-1)), 0)))
    end:
A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))):
seq(seq(A(n, d-n), n=0..d), d=0..9);

b[args__] := b[args] = With[{n = Length[{args}]}, If[n<2 || {0} == Union[ {args}], 1, If[n==2, If[{args}[[1]]>0, b[{args}[[1]]-1, {args}[[2]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, {args}[[1]] ], 0], If[{args}[[1]]>0, b[{args}[[1]]-1, Sequence @@ {args}[[2;;n]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, Sequence @@ {args}[[3;;n]], {args}[[1]] ], 0]+ If[{args}[[n]]>0, b[{args}[[n]]-1, Sequence @@ Most[{args}]] ],0]] /. Null -> 0];
a[n_,k_]:= If[n==0 || k==0, 1, b[n-1, Sequence @@ Array[n&, k-1]]];
Table[Table[a[n, d-n], {n,0,d}], {d,0,9}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A208674 Number of words, either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs 3 times and letters of neighboring word positions are equal or neighbors in the alphabet.

Original entry on oeis.org

1, 1, 10, 37, 163, 640, 2503, 9559, 36154, 135541, 505279, 1875592, 6941035, 25629211, 94478338, 347857921, 1279622611, 4704064120, 17284247263, 63484653151, 233114307274, 855817783741, 3141437229271, 11529935743528, 42314502514051, 155283277278547

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (3*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (3,3,...,3) with positive unit steps in all dimensions such that the absolute difference of the dimension indices used in consecutive steps is <= 1.

			a(2) = 10 = |{aaabbb, aababb, aabbab, aabbba, abaabb, ababab, ababba, abbaab, abbaba, abbbaa}| with binary alphabet {a,b}.
a(3) = 37 = |{aaabbbccc, aaabbcbcc, aaabbccbc, aaabbcccb, aaabcbbcc, aaabcbcbc, aaabcbccb, aaabccbbc, aaabccbcb, aaabcccbb, aababbccc, aababcbcc, aababccbc, aababcccb, aabbabccc, aabbcccba, aabcbabcc, aabcbccba, aabccbabc, aabccbcba, aabcccbab, aabcccbba, abaabbccc, abaabcbcc, abaabccbc, abaabcccb, abababccc, ababcccba, abbaabccc, abbcccbaa, abcbaabcc, abcbccbaa, abccbaabc, abccbcbaa, abcccbaab, abcccbaba, abcccbbaa}| with ternary alphabet {a,b,c}.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-1,-14,-2,4)

Row n=3 of A208673.

a:= n-> (Matrix(5, (i, j)-> `if`(i=j-1, 1, `if`(i=5, [4, -2,
        -14, -1, 5][j], 0)))^n. <<1, 1, 10, 37, 163>>)[1, 1]:
seq(a(n), n=0..30);

G.f.: -(4*x^4+2*x^3+6*x^2-4*x+1) / (4*x^5-2*x^4-14*x^3-x^2+5*x-1).

A351759 Number of words either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

Original entry on oeis.org

1, 1, 3, 37, 3593, 2336376, 12085125703, 492300320300251, 163856834420168837331, 450436951076447377275760495, 10360995769563446558192576106611177, 2010994252878923176109086647579398800496256, 3317129856969862808949985510559979253492213624483355

Offset: 0

Alois P. Heinz, Feb 18 2022

nonn

Also the number of (n^2-1)-step walks on n-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.

			a(2) = 3:
  +----+   +----+   +----+
  |aabb|   |abab|   |abba|
  +----+   +----+   +----+
  |1222|   |1122|   |1112|
  |0012|   |0112|   |0122|
  +----+   +----+   +----+
  |xx  |   |x x |   |x  x|
  |  xx|   | x x|   | xx |
  +----+   +----+   +----+
a(3) = 37: aaabbbccc, aaabbcbcc, aaabbccbc, aaabbcccb, aaabcbbcc, aaabcbcbc, aaabcbccb, aaabccbbc, aaabccbcb, aaabcccbb, aababbccc, aababcbcc, aababccbc, aababcccb, aabbabccc, aabbcccba, aabcbabcc, aabcbccba, aabccbabc, aabccbcba, aabcccbab, aabcccbba, abaabbccc, abaabcbcc, abaabccbc, abaabcccb, abababccc, ababcccba, abbaabccc, abbcccbaa, abcbaabcc, abcbccbaa, abccbaabc, abccbcbaa, abcccbaab, abcccbaba, abcccbbaa.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Main diagonal of A208673.

b:= proc(l, t) option remember; (n-> `if`(l=[0$n], 1,
      add(`if`(l[i]=0, 0, b(subsop(i=l[i]-1, l), i)),
           i=max(1, t-1)..min(n, t+1))))(nops(l))
    end:
a:= n-> b([n$n], 0):
seq(a(n), n=0..6);

b[l_, t_] := b[l, t] = Function [n, If[l == Array[0&, n], 1,
     Sum[If[l[[i]] == 0, 0, b[ReplacePart[l, i -> l[[i]] - 1], i]],
     {i, Max[1, t - 1], Min[n, t + 1]}]]][Length[l]];
a[n_] := b[Array[n&, n], 0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

a(n) = A208673(n,n).

cp's OEIS Frontend

A212334 Number of words, either empty or beginning with the first letter of the 4-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

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A208675 Number of words, either empty or beginning with the first letter of the ternary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

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A208879 Number of words A(n,k), either empty or beginning with the first letter of the cyclic 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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A208674 Number of words, either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs 3 times and letters of neighboring word positions are equal or neighbors in the alphabet.

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A351759 Number of words either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

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