A081908 a(n) = 2^n*(n^2 - n + 8)/8.

1, 2, 5, 14, 40, 112, 304, 800, 2048, 5120, 12544, 30208, 71680, 167936, 389120, 892928, 2031616, 4587520, 10289152, 22937600, 50855936, 112197632, 246415360, 538968064, 1174405120, 2550136832, 5519704064, 11911823360, 25635586048

Offset: 0

Author

Paul Barry, Mar 31 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A000124 (when this begins 1,1,2,4,7,...).

2nd binomial transform of (1,0,1,0,0,0,...).

Case k=2 where a(n,k) = k^n(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A081909.

Cf. A000124, A000079, A001788, A001787.

Programs

Magma

[2^n*(n^2-n+8)/8: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

Mathematica

Table[2^n*(n^2-n+8)/8, {n,0,50}] (* or *) LinearRecurrence[{6,-12,8}, {1, 2,5}, 50] (* G. C. Greubel, Oct 17 2018 *)

PARI

a(n)=2^n*(n^2-n+8)/8 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1 - 4*x + 5*x^2)/(1-2*x)^3.

a(n) = A000079(n) + (A001788(n) - A001787(n))/2. - Paul Barry, May 27 2003

a(n) = Sum_{k=0..n} C(n, k)*(1 + C(k, 2)). - Paul Barry, May 27 2003

E.g.f.: (2 + x^2)*exp(2*x)/2. - G. C. Greubel, Oct 17 2018