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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

a(n) ~ 2^(8*n + 7/2) / (sqrt(Pi*n) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Oct 19 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+3,n-2*k). - Seiichi Manyama, Nov 11 2025

Mathematica

Table[Sum[Binomial[4*n+2, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 16 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n+2, k));
(Magma) [&+[Binomial(4*n+2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Crossrefs

Cf. A066381, A386811, A387010, A387011.

Cf. A000346, A387007.

Cf. A002293.

Sequence in context: A147839 A126694 A390592 * A305198 A264912 A323216

Adjacent sequences: A387006 A387007 A387008 * A387010 A387011 A387012

Keyword

nonn

Author

Seiichi Manyama, Aug 12 2025

Status

approved