A007744 Expansion of (1+6*x)/(1-4*x)^(7/2).

1, 20, 210, 1680, 11550, 72072, 420420, 2333760, 12471030, 64664600, 327202876, 1622493600, 7909656300, 38003792400, 180324117000, 846321189120, 3934071152550, 18132120329400, 82937661506700

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Fourth column in A104684. - Paul Barry, May 02 2005

Diagonal of the rational function 1 / (1 - x - y)^4. - Ilya Gutkovskiy, Apr 23 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
• A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013.
• G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

Programs

Magma

[Binomial(2*n+3, n)*Binomial(n+3, 3): n in [0..20]]; // Vincenzo Librandi, Aug 20 2011

Mathematica

Array[Binomial[2 # + 3, #]*Binomial[# + 3, 3] &, 19, 0] (* Michael De Vlieger, Aug 18 2021 *)

Formula

a(n) = binomial(2n+3, n) * binomial(n+3, 3). - Paul Barry, May 02 2005

G.f.: G(0) where G(k) = 1 + 4*x*(k+1)*(4*k+5)/((2*k+1)^2 - x*(2*k+1)^2*(2*k+3)*(4*k+7)/(x*(2*k+3)*(4*k+7) + 2*(k+1)^2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 12 2012

D-finite with recurrence: n*a(n) + 2*(n-11)*a(n-1) + 12*(-2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012