A387009 a(n) = Sum_{k=0..n} binomial(4*n+2,k).
1, 7, 56, 470, 4048, 35443, 313912, 2804012, 25211936, 227881004, 2068564064, 18844224462, 172186125456, 1577401391626, 14483100716176, 133240186921816, 1227901991526976, 11333497984085620, 104752914242685856, 969417048912326008, 8981452266787224128
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[Binomial(4*n+2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025
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Mathematica
Table[Sum[Binomial[4*n+2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n+2, k));
Formula
a(n) = [x^n] (1+x)^(4*n+2)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n+2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+2,k) * binomial(4*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+1,n-k).
G.f.: g^3/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-3)*(2*n-1)*(4*n-5)*(22*n+5)*a(n-2) -8*(1892*n^4-3706*n^3+1750*n^2+214*n-177)*a(n-1) +3*(22*n-17)*(n-1)*(3*n-1)*(3*n+1)*a(n) = 0. - Georg Fischer, Aug 17 2025