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%I #39 Nov 24 2022 16:21:45
%S 1,3,9,19,39,69,119,189,294,434,630,882,1218,1638,2178,2838,3663,4653,
%T 5863,7293,9009,11011,13377,16107,19292,22932,27132,31892,37332,43452,
%U 50388,58140,66861,76551,87381,99351,112651,127281
%N G.f.: 1/((1-x)*(1-x^2))^3.
%C 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
%C Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10, ...]. - _Gary W. Adamson_, Jun 11 2009
%C 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
%C 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
%H Alois P. Heinz, <a href="/A038163/b038163.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-8,6,6,-8,0,3,-1).
%F 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.
%F 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
%F a(n+1) - a(n) = A096338(n+2). - _R. J. Mathar_, Nov 04 2008
%p 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
%p # alternative
%p A038163 := proc(n)
%p (4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555))/3840 ;
%p end proc:
%p seq(A038163(n),n=0..30) ; # _R. J. Mathar_, Feb 22 2021
%t CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* _Jean-François Alcover_, Mar 11 2014 *)
%t LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1},{1,3,9,19,39,69,119,189,294},50] (* _Harvey P. Dale_, Nov 24 2022 *)
%o (Haskell)
%o import Data.List (inits, intersperse)
%o a038163 n = a038163_list !! n
%o a038163_list = map
%o (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $
%o tail $ inits $ tail a000217_list where
%o -- _Reinhard Zumkeller_, Feb 27 2015
%Y Cf. A008619, A006918, A001753.
%Y Cf. A096338.
%Y Column k=3 of A210391. - _Alois P. Heinz_, Mar 22 2012
%Y Cf. A000217.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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