login
a(n) = binomial(12*n, n).
10

%I #29 Sep 08 2022 08:45:49

%S 1,12,276,7140,194580,5461512,156238908,4529365776,132601016340,

%T 3911395881900,116068178638776,3461014728350400,103619293824707388,

%U 3112781199432937200,93780365051563029360,2832430653037446854640,85733828145510955528212,2600022926684976508835280

%N a(n) = binomial(12*n, n).

%H Vincenzo Librandi, <a href="/A169961/b169961.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = C(12*n-1,n-1)*C(144*n^2,2)/(3*n*C(12*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014

%F From _Bradley Klee_, Jul 01 2018 : (Start)

%F G.f. G(x) and derivatives G^(n)(x)=d^n/dx^n G(x) satisfy a Picard-Fuchs type differential equation, 0=Sum_{m=0..11}(v1_{n}*x^(n+1)-v2_{n}*x^n)*G^(n)(x), with integer coefficient vectors:

%F v1={479001600, 647647046323200, 99278289544896000, 1290870365178240000, 4245175263164774400, 5313701967430348800, 3083267876011868160, 918801061774295040, 147161631039160320, 12624021804810240, 539424077119488, 8916100448256}

%F v2={0, 39916800, 14079254112000, 1273481816745600, 11475123393888000, 27687351298068000, 25909403608075680, 11200182937408080, 2427742942653600, 268452344620350, 14265583530550, 285311670611}

%F G.f.: G(x) = 11F10(m/12;n/11;12^12/11^11*x), m=1..11, n=1..10. (End)

%F From _Vaclav Kotesovec_, Jul 15 2018: (Start)

%F Recurrence: 11*n*(11*n - 10)*(11*n - 9)*(11*n - 8)*(11*n - 7)*(11*n - 6)*(11*n - 5)*(11*n - 4)*(11*n - 3)*(11*n - 2)*(11*n - 1)*a(n) = 41472*(2*n - 1)*(3*n - 2)*(3*n - 1)*(4*n - 3)*(4*n - 1)*(6*n - 5)*(6*n - 1)*(12*n - 11)*(12*n - 7)*(12*n - 5)*(12*n - 1)*a(n-1).

%F a(n) ~ 2^(24*n + 1/2) * 3^(12*n + 1/2) / (sqrt(Pi*n) * 11^(11*n + 1/2)). (End)

%F From _Peter Bala_, Feb 21 2022: (Start)

%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 11*A(x))^11 + (12^12)*x*A(x)^12 = 0.

%F Sum_{n >= 1} a(n)*( x*(11*x + 12)^11/(12^12*(1 + x)^12) )^n = x. (End)

%t Table[Binomial[12 n, n], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 07 2014 *)

%t CoefficientList[Series[HypergeometricPFQ[Range[11]/12, Range[10]/11,(12^12)/(11^11)*x], {x,0,10}],x] (* _Bradley Klee_, Jul 01 2018 *)

%o (Magma) [Binomial(12*n, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014

%o (PARI) a(n) = binomial(12*n, n); \\ _Michel Marcus_, Jul 02 2018

%Y Cf. A000984, A005809, A005810, A001449, A004355, A004368, A004381, A169958, A169959, A169960.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 07 2010