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

0,2

Links

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

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

a(n) ~ 2^(8*n - 13/2) / (sqrt(Pi*n) * 3^(3*n - 7/2)). - Vaclav Kotesovec, Oct 19 2025

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

Mathematica

Table[Sum[Binomial[4*n-3, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n-3, k));
(Magma) [&+[Binomial(4*n-3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387036, A387037.

Cf. A002293, A262977.

Sequence in context: A360321 A074623 A323946 * A275636 A002576 A037727

Adjacent sequences: A387032 A387033 A387034 * A387036 A387037 A387038

Keyword

nonn

Author

Seiichi Manyama, Aug 13 2025

Status

approved