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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 Alois P. Heinz, Table of n, a(n) for n = 0..250 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 Adjacent sequences: A141054 A141055 A141056 * A141058 A141059 A141060 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.)