A079727 a(n) = 1 + C(2,1)^3 + C(4,2)^3 + ... + C(2n,n)^3.

1, 9, 225, 8225, 351225, 16354233, 805243257, 41229480825, 2172976383825, 117106008311825, 6423711336265041, 357470875526646609, 20131502573232075025, 1145190201805448075025, 65706503254247744075025

Offset

0,2

Comments

a(n) seems to have an interesting congruence property: For p prime, a(p)==8 (mod p) if and only if p == 3, 5, 7, or 13 (mod 14); i.e., iff p=7 or p is in A003625.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..556

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k)^3.

G.f.: hypergeom([1/2, 1/2, 1/2], [1, 1], 64*x)/(1-x). - Vladeta Jovovic, Feb 18 2003

G.f.: hypergeom([1/4,1/4],[1],64*x)^2/(1-x). - Mark van Hoeij, Nov 17 2011

Recurrence: (n+2)^3*a(n+2)-(5*n+8)*(13*n^2+38*n+28)*a(n+1)+8*(2n+3)^3*a(n)=0. - Emanuele Munarini, Nov 15 2016

a(n) ~ 2^(6*n+6) / (63*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 16 2016

Mathematica

Table[Sum[Binomial[2 k, k]^3, {k, 0, n}], {n, 0, 14}] (* Michael De Vlieger, Nov 15 2016 *)

Prog

(PARI) a(n)=sum(k=0, n, binomial(2*k, k)^3)
(Maxima) makelist(sum(binomial(2*k, k)^3, k, 0, n), n, 0, 12); /* Emanuele Munarini, Nov 15 2016 */
(Magma) [&+[Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 16 2016

Crossrefs

Cf. A002476.

Cf. Sum_{k = 0..n} binomial(2*k, k)^m: A006134 (m=1), A115257 (m=2), this sequence (m=3).

Sequence in context: A012831 A012749 A188662 * A251579 A128492 A294971

Adjacent sequences: A079724 A079725 A079726 * A079728 A079729 A079730

Keyword

nonn

Author

Benoit Cloitre, Feb 17 2003

Status

approved