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

0,2

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).

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)

Mathematica

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

Prog

(Magma) [2^(n-2)*n*(5+n) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011
(PARI) a(n)=n*(n+5)<<(n-2) \\ Charles R Greathouse IV, Sep 21 2017

Crossrefs

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

Sequence in context: A006900 A212505 A024453 * A345690 A139263 A354507

Adjacent sequences: A176024 A176025 A176026 * A176028 A176029 A176030

Keyword

nonn,easy

Author

Paul Curtz, Dec 06 2010

Status

approved