A075156 Binomial transform of pentanacci numbers A074048: a(n) = Sum_{k=0..n} binomial(n,k)*A074048(k).

5, 6, 10, 24, 70, 216, 664, 2008, 5998, 17808, 52770, 156360, 463492, 1374392, 4076222, 12090144, 35859742, 106359928, 315460168, 935639768, 2775057510, 8230670416, 24411730298, 72403913480, 214746249796, 636926269816

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 07 2002

Keywords

• easy
• nonn

Links

• N. J. A. Sloane, Transforms
• Index entries for linear recurrences with constant coefficients, signature (6,-13,14,-7,2).

Crossrefs

Cf. A074048.

Programs

Maple

M := Matrix(5, (i,j)-> if (i=j-1) then 1 elif j>1 then 0 else [6,-13,14,-7,2][i] fi); a := n -> (Matrix([[70,24,10,6,5]]).M^(n))[1,5]; seq (a(n), n=0..50); # Alois P. Heinz, Jul 25 2008

Mathematica

CoefficientList[Series[(5-24*x+39*x^2-28*x^3+7*x^4)/(1-6*x+13*x^2-14*x^3+7*x^4-2*x^5), {x, 0, 25}], x]
LinearRecurrence[{6,-13,14,-7,2},{5,6,10,24,70},30] (* Harvey P. Dale, Mar 10 2019 *)

Formula

a(n) = 6a(n-1) - 13a(n-2) + 14a(n-3) - 7a(n-4) + 2a(n-5), a(0)=5, a(1)=6, a(2)=10, a(3)=24, a(4)=70.

G.f.: (5 - 24*x + 39*x^2 - 28*x^3 + 7*x^4)/(1 - 6*x + 13*x^2 - 14*x^3 + 7*x^4 - 2*x^5).

a(n) = term (1,5) in the 1 X 5 matrix [70,24,10,6,5]. [6,1,0,0,0; -13,0,1,0,0; 14,0,0,1,0; -7,0,0,0,1; 2,0,0,0,0]^n. - Alois P. Heinz, Jul 25 2008