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.
1, 1, 5, 37, 309, 2751, 25493, 242845, 2360501, 23301307, 232834755, 2349638259, 23905438725, 244889453043, 2523373849701, 26132595017037, 271826326839477, 2838429951771795, 29740725671232119, 312573076392760183, 3294144659048391059, 34802392680979707121
Offset: 0
Examples
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}|.
Links
- 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.
Programs
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Magma
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
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Maple
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
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Mathematica
a[n_]:= HypergeometricPFQ[{1-n,-n,n}, {1,1}, 1] (* Michael Somos, Jun 03 2012 *) -
SageMath
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
Formula
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) = [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)
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