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A141057 Number of Abelian cubes of length 3n over an alphabet of size 3. An Abelian cube is a string of the form x x' x'' with |x| = |x'| = |x''| and x is a permutation of x' and x''. 6
1, 3, 27, 381, 6219, 111753, 2151549, 43497891, 912018123, 19671397617, 434005899777, 9754118112951, 222621127928109, 5147503311510927, 120355825553777043, 2841378806367492381, 67648182142185172683, 1622612550613755130497, 39178199253650491044441 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for primes p >= 5 and positive integers n and k. Extending the sequence to negative n via a(-n) = Sum_{k = 0..n} C(-n,k)^3 * Sum_{j = 0..k} C(k,j)^3 produces the sequence [-1, 255, -53893, 14396623, -4388536251, 1461954981315, -518606406878589, ...] that appears to satisfy the same supercongruences. - Peter Bala, Apr 27 2022
LINKS
FORMULA
a(n) = sum of (n!/(n1)! (n2)! (n3!))^3 over all nonnegative n1, n2, n3 such that n1+n2+n3 = n.
G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = [ Sum_{n>=0} x^n/n!^3 ]^3. - Paul D. Hanna, Jan 19 2011
a(n) = Sum_{k=0..n} C(n,k)^3 * Sum_{j=0..k} C(k,j)^3 = Sum_{k=0..n} C(n,k)^3*A000172(k). - Paul D. Hanna, Jan 20 2011
a(n) ~ 3^(3*n+2) / (4 * Pi^2 * n^2). - Vaclav Kotesovec, Sep 04 2014
a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^3. - Peter Luschny, May 31 2017
EXAMPLE
a(1) = 3 as the Abelian cubes are aaa, bbb, ccc.
G.f.: A(x) = 1 + 3*x + 27*x^2/2!^3 + 381*x^3/3!^3 + 6219*x^4/4!^3 +...
A(x) = [1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +...]^3. - Paul D. Hanna
MAPLE
a:= proc(n) option remember; `if`(n<3, [1, 3, 27][n+1],
((567*n^6-3213*n^5+7083*n^4-7920*n^3+4968*n^2-1680*n+240)*a(n-1)
-3*(3*n-4)*(63*n^5-399*n^4+1039*n^3-1380*n^2+920*n-240)*a(n-2)
+729*(21*n^2-35*n+15)*(n-2)^4*a(n-3))/(n^4*(21*n^2-77*n+71)))
end:
seq(a(n), n=0..20); # Alois P. Heinz, May 25 2013
A141057_list := proc(len) series(hypergeom([], [1, 1], x)^3, x, len);
seq((n!)^3*coeff(%, x, n), n=0..len-1) end:
A141057_list(19); # Peter Luschny, May 31 2017
MATHEMATICA
a[n_] := Sum[Binomial[n, k]^3 HypergeometricPFQ[{-k, -k, -k}, {1, 1}, -1], {k, 0, n}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jun 27 2019 *)
PROG
(PARI) {a(n)=if(n<0, 0, n!^3*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^3, n))}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^3*sum(j=0, k, binomial(k, j)^3))}
(PARI) N=33; x='x+O('x^N)
Vec(serlaplace(serlaplace(serlaplace(sum(n=0, N, x^n/(n!^3)))^3))) /* show terms */
CROSSREFS
Cf. A000172 (Franel numbers), A002893.
Sequence in context: A157089 A365794 A138436 * A365569 A365586 A201696
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Aug 01 2008
EXTENSIONS
Extended by Paul D. Hanna, Jan 19 2011
Offset corrected by Alois P. Heinz, May 25 2013
STATUS
approved

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Last modified May 27 11:22 EDT 2024. Contains 372858 sequences. (Running on oeis4.)