A186264 Expansion of 3F2( 1, 3/2, 3/2; 3, 4;16 x).

1, 3, 15, 98, 756, 6534, 61347, 613470, 6447012, 70526404, 797490876, 9271926888, 110380082000, 1341117996300, 16586474042475, 208360804638150, 2653858669601700, 34220809160653500, 446174168961282300, 5875592302944678600, 78078028942687784400

Offset: 0

Author

Olivier Gérard, Feb 16 2011

Keywords

• nonn
• easy

Comments

Combinatorial interpretation welcome.

Links

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

Crossrefs

Cf. A186262, A000108, A001246.

Programs

Maple

seq(3/((n+3)*(n+2)^2)*binomial(2*n+2,n+1)^2, n = 0..20); # Peter Bala, Mar 28 2018

Mathematica

CoefficientList[Series[HypergeometricPFQ[{1, 3/2, 3/2}, {3, 4}, 16*x], {x, 0, 20}],
  x]

Formula

G.f. is equivalent to -3*( 1+2*x -2F1(-1/2,-1/2;2;16*x) ) /(4*x^2).

a(n) = 3/((n+3)*(n+2)^2)*(2*n+2)!^2/(n+1)!^4 = 3/(n+3)* Catalan(n+1)^2. - Peter Bala, Mar 28 2018

D-finite with recurrence (n+3)*(n+2)*a(n) -4*(2*n+1)^2*a(n-1)=0. - R. J. Mathar, Feb 08 2021