A169961 a(n) = binomial(12*n, n).

1, 12, 276, 7140, 194580, 5461512, 156238908, 4529365776, 132601016340, 3911395881900, 116068178638776, 3461014728350400, 103619293824707388, 3112781199432937200, 93780365051563029360, 2832430653037446854640, 85733828145510955528212, 2600022926684976508835280

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

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

From Bradley Klee, Jul 01 2018 : (Start)

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:

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

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

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

From Vaclav Kotesovec, Jul 15 2018: (Start)

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).

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

From Peter Bala, Feb 21 2022: (Start)

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

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

Mathematica

Table[Binomial[12 n, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)
CoefficientList[Series[HypergeometricPFQ[Range[11]/12, Range[10]/11, (12^12)/(11^11)*x], {x, 0, 10}], x] (* Bradley Klee, Jul 01 2018 *)

Prog

(Magma) [Binomial(12*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
(PARI) a(n) = binomial(12*n, n); \\ Michel Marcus, Jul 02 2018

Crossrefs

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

Sequence in context: A117415 A152129 A115461 * A363384 A220383 A296383

Adjacent sequences: A169958 A169959 A169960 * A169962 A169963 A169964

Keyword

nonn

Author

N. J. A. Sloane, Aug 07 2010

Status

approved