cp's OEIS Frontend

a(n) = C(9*n-1, n-1)*C(81*n^2, 2)/(3*n*C(9*n+1, 3)), n > 0. - Gary Detlefs, Jan 02 2014

From Peter Bala, Feb 21 2022: (Start)

The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 8*A(x))^8 + (9^9)*x*A(x)^9 = 0.

Sum_{n >= 1} a(n)*( x*(8*x + 9)^8/(9^9*(1 + x)^9) )^n = x. (End)

D-finite with recurrence 128*n*(8*n-5) *(4*n-1) *(8*n-7) *(2*n-1) *(8*n-1) *(4*n-3) *(8*n-3)*a(n) -81*(9*n-7) *(9*n-5) *(3*n-1) *(9*n-1) *(9*n-8) *(3*n-2) *(9*n-4) *(9*n-2)*a(n-1)=0. - R. J. Mathar, Aug 19 2025

G.f.: 8F7(8/9, 7/9, 2/3, 5/9, 4/9, 1/3, 2/9 ,1/9 ; 7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8; 387420489/16777216*x). - R. J. Mathar, Aug 19 2025

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)

From Seiichi Manyama, Aug 16 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(12*n+1,k).

G.f.: 1/(1 - 12*x*g^11) where g = 1+x*g^12.

G.f.: g/(12-11*g) where g = 1+x*g^12. (End)