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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''. 3
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

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-Fran├žois 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).

Sequence in context: A328182 A157089 A138436 * A201696 A011781 A094577

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 November 12 07:00 EST 2019. Contains 329052 sequences. (Running on oeis4.)