A387035 a(n) = Sum_{k=0..n} binomial(4*n-3,k).
1, 2, 16, 130, 1093, 9402, 82160, 726206, 6474541, 58115146, 524472448, 4754293704, 43257431931, 394821713910, 3613377083248, 33146854168628, 304692552429413, 2805871076597738, 25880523571338272, 239058748663208600, 2211058130414688244, 20474163633488699944
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Magma
[&+[Binomial(4*n-3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[Binomial[4*n-3,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n-3, k));
Formula
a(n) = [x^n] (1+x)^(4*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-3,k) * binomial(4*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-7)*(2*n-5)*(4*n-9)*(22*n^3-50*n^2+5*n+30)*a(n-2) -8*(1892*n^6-16004*n^5+51038*n^4-73470*n^3+39874*n^2+6165*n-9450)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^3-116*n^2+171*n-47)*a(n) = 0. - Georg Fischer, Aug 17 2025