A176027 Binomial transform of A005563.

0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, 479232, 1089536, 2457600, 5505024, 12255232, 27131904, 59768832, 131072000, 286261248, 622854144, 1350565888, 2919235584, 6291456000, 13522436096

Offset: 0

Author

Paul Curtz, Dec 06 2010

Keywords

• nonn
• easy

Comments

The numbers appear on the diagonal of a table T(n,k), where the left column contains the elements of A005563, and further columns are recursively T(n,k) = T(n,k-1)+T(n-1,k-1):

....0....-1.....0.....0.....0.....0.....0.....0.....0.....0.

....3.....3.....2.....2.....2.....2.....2.....2.....2.....2.

....8....11....14....16....18....20....22....24....26....28.

...15....23....34....48....64....82...102...124...148...174.

...24....39....62....96...144...208...290...392...516...664.

...35....59....98...160...256...400...608...898..1290..1806.

...48....83...142...240...400...656..1056..1664..2562..3852.

...63...111...194...336...576...976..1632..2688..4352..6914.

...80...143...254...448...784..1360..2336..3968..6656.11008.

...99...179...322...576..1024..1808..3168..5504..9472.16128.

..120...219...398...720..1296..2320..4128..7296.12800.22272.

The second column is A142463, the third A060626, the fourth essentially A035008 and the fifth essentially A016802. Transposing the array gives A005563 and its higher order differences in the individual rows.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A001792, A001793, A005563, A058396, A084266, A127276.

Programs

Magma

[2^(n-2)*n*(5+n) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Mathematica

LinearRecurrence[{6,-12,8},{0,3,14},30] (* Harvey P. Dale, Oct 19 2015 *)

PARI

a(n)=n*(n+5)<<(n-2) \\ Charles R Greathouse IV, Sep 21 2017

Formula

G.f.: x*(-3+4*x)/(2*x-1)^3. - R. J. Mathar, Dec 11 2010

a(n) = 2^(n-2)*n*(5+n). - R. J. Mathar, Dec 11 2010

a(n) = A127276(n) - A127276(n+1).

a(n+1)-a(n) = A084266(n+1).

a(n+2) = 16*A058396(n) for n > 0.

a(n) = 2*a(n-1) + A001792(n).

a(n) = A001793(n) - 2^(n-1) for n > 0. - Brad Clardy, Mar 02 2012

a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+3) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017

From Amiram Eldar, Aug 13 2022: (Start)

Sum_{n>=1} 1/a(n) = 1322/75 - 124*log(2)/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = 132*log(3/2)/5 - 782/75. (End)