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A038163 G.f.: 1/((1-x)*(1-x^2))^3. 17
1, 3, 9, 19, 39, 69, 119, 189, 294, 434, 630, 882, 1218, 1638, 2178, 2838, 3663, 4653, 5863, 7293, 9009, 11011, 13377, 16107, 19292, 22932, 27132, 31892, 37332, 43452, 50388, 58140, 66861, 76551, 87381, 99351, 112651, 127281 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of symmetric nonnegative integer 6 X 6 matrices with sum of elements equal to 4*n, under action of dihedral group D_4. - Vladeta Jovovic, May 14 2000

Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10,...]. - Gary W. Adamson, Jun 11 2009

Number of partitions of n (n>=1) into 1s and 2s if there are three kinds of 1s and three kinds of 2s. Example: a(2)=9 because we have 11, 11', 11", 1'1', 1'1", 1"1", 2, 2', and 2". - Emeric Deutsch, Jun 26 2009

Equals the tetrahedral numbers with repeats convolved with the natural numbers: (1 + x + 4x^2 + 4x^3 + ...) * (1 + 2x + 3x^2 + 4x^3 + ...) = (1 + 3x + 9x^2 + 19x^3 + ...). - Gary W. Adamson, Dec 22 2010

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

FORMULA

a(2*k) = (4*k + 5)*binomial(k + 4, 4)/5 = A034263(k); a(2*k + 1) = binomial(k + 4, 4)*(15 + 4*k)/5 = A059599(k), k >= 0.

a(n) = 1/3840*(4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555)). Recurrence: a(n) = 3*a(n-1)-8*a(n-3)+6*a(n-4)+6*a(n-5)-8*a(n-6)+3*a(n-8)-a(n-9). - Vladeta Jovovic, Apr 24 2002

a(n+1)-a(n) = A096338(n+2). - R. J. Mathar, Nov 04 2008

MAPLE

G := 1/((1-x)^3*(1-x^2)^3): Gser := series(G, x = 0, 42): seq(coeff(Gser, x, n), n = 0 .. 37); # Emeric Deutsch, Jun 26 2009

MATHEMATICA

CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* Jean-Fran├žois Alcover, Mar 11 2014 *)

PROG

(Haskell)

import Data.List (inits, intersperse)

a038163 n = a038163_list !! n

a038163_list = map

    (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $

    tail $ inits $ tail a000217_list where

-- Reinhard Zumkeller, Feb 27 2015

CROSSREFS

Cf. A008619, A006918, A001753.

Cf. A096338.

Column k=3 of A210391. - Alois P. Heinz, Mar 22 2012

Cf. A000217.

Sequence in context: A005994 A080010 A135117 * A146819 A147213 A146441

Adjacent sequences:  A038160 A038161 A038162 * A038164 A038165 A038166

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 16 08:40 EDT 2018. Contains 316259 sequences. (Running on oeis4.)