A081187 5th binomial transform of (1,0,1,0,1,...), A059841.

1, 5, 26, 140, 776, 4400, 25376, 148160, 872576, 5169920, 30757376, 183495680, 1096779776, 6563901440, 39316299776, 235629363200, 1412702437376, 8471919656960, 50814338072576, 304817308958720, 1828628975845376

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A081187.

a(n) is also the number of words of length n over an alphabet of six letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs, for the 1- to 11-letter word cases. - Wolfdieter Lang, Jul 17 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (10,-24).

Crossrefs

Cf. A007582, A074612, A081189.

Programs

GAP

List([0..25], n-> (4^n + 6^n)/2); # G. C. Greubel, Dec 26 2019

Magma

[4^n/2 + 6^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Maple

seq( (4^n + 6^n)/2, n=0..25); # G. C. Greubel, Dec 26 2019

Mathematica

CoefficientList[Series[(1-5x)/((1-4x)(1-6x)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{10,-24}, {1,5}, 26] (* G. C. Greubel, Dec 26 2019 *)

PARI

vector(26, n, (4^(n-1) + 6^(n-1))/2) \\ G. C. Greubel, Dec 26 2019

Sage

[(4^n + 6^n)/2 for n in (0..25)] # G. C. Greubel, Dec 26 2019

Formula

a(n) = 10*a(n-1) - 24*a(n-2) with n > 1, a(0)=1, a(1)=5.

G.f.: (1-5*x)/((1-4*x)*(1-6*x)).

E.g.f.: exp(5*x)*cosh(x).

a(n) = (4^n + 6^n)/2.

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*5^(n-2k).

E.g.f.: exp(5*x)*cosh(x) = (1/2)*E(0), where E(k) = 1 + 2^k/(3^k - 6*x*9^k/(6*x*3^k + (k+1)*2^k/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011

a(n) = A074612(n)/2. - G. C. Greubel, Jan 13 2024