A080961 Fourth binomial transform of A010686 (period 2: repeat 1,5).

1, 9, 57, 321, 1713, 8889, 45417, 230001, 1158753, 5820009, 29178777, 146130081, 731358993, 3658920729, 18300980937, 91524036561, 457677578433, 2288560079049, 11443316955897, 57218134461441, 286095321353073, 1430490553902969, 7152494610927657, 35762598578876721

Offset: 0

Author

Paul Barry, Mar 03 2003

Keywords

• easy
• nonn

Examples

			G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-15).

Crossrefs

Cf. A003948, A005059, A010686, A080960, A080962.

Programs

Magma

binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;
binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // Klaus Brockhaus, Nov 26 2009

Magma

[3*5^n - 2*3^n: n in [0..30]]; // Vincenzo Librandi, Dec 07 2012

Maple

A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2016

Mathematica

CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 07 2012 *)

Formula

a(n) = 5*a(n-1) + 4*3^(n-1).

a(n) = 3*5^n - 2*3^n.

G.f.: (1+x)/((1-3*x)*(1-5*x)). - Klaus Brockhaus, Nov 26 2009

From Mario C. Enriquez, Dec 08 2016: (Start)

a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.

a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)

a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - Wesley Ivan Hurt, Dec 08 2016

E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 2024

Extensions

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009