A081144 Starting at 1, four-fold convolution of A000400 (powers of 6).

0, 0, 0, 1, 24, 360, 4320, 45360, 435456, 3919104, 33592320, 277136640, 2217093120, 17293326336, 132058128384, 990435962880, 7313988648960, 53287631585280, 383670947414016, 2733655500324864, 19296391766999040, 135074742368993280

Offset

0,5

Comments

With a different offset, number of n-permutations (n=4) of 7 objects: t, u, v, w, z, x, y with repetition allowed, containing exactly three u's. Example: a(4)=24 because we have uuut, uutu, utuu, tuuu, uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu, xuuu, uuuy, uuyu, uyuu, yuuu. - Zerinvary Lajos, Jun 03 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (24,-216,864,-1296).

Formula

G.f.: x^3/(1 - 6*x)^4.

a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4) for n > 3, a(0) = a(1) = a(2) = 0, a(3) = 1.

a(n) = 6^(n - 3)*binomial(n, 3).

From Amiram Eldar, Jan 04 2022: (Start)

Sum_{n>=3} 1/a(n) = 450*log(6/5) - 81.

Sum_{n>=3} (-1)^(n+1)/a(n) = 882*log(7/6) - 135. (End)

Maple

seq(seq(binomial(i+2, j)*6^(i-1), j =i-1), i=-2..19); # Zerinvary Lajos, Dec 30 2007
seq(binomial(n+3, 3)*6^n, n=-3..18); # Zerinvary Lajos, Jun 03 2008

Prog

(Sage)[lucas_number2(n, 6, 0)*binomial(n, 3)/6^3 for n in range(0, 22)] # Zerinvary Lajos, Mar 13 2009
(Magma) [6^n* Binomial(n+3, 3): n in [-3..20]]; // Vincenzo Librandi, Oct 16 2011
(GAP) List([-3..18], n->Binomial(n+3, 3)*6^n); # Muniru A Asiru, Feb 19 2018

Crossrefs

Cf. A081143, A081136.

Cf. A038255.

Sequence in context: A028245 A005546 A309843 * A126780 A052741 A046905

Adjacent sequences: A081141 A081142 A081143 * A081145 A081146 A081147

Keyword

nonn,easy

Author

Paul Barry, Mar 08 2003

Status

approved