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%I #39 Feb 05 2022 15:45:31
%S 1,4,12,28,58,108,188,308,483,728,1064,1512,2100,2856,3816,5016,6501,
%T 8316,10516,13156,16302,20020,24388,29484,35399,42224,50064,59024,
%U 69224,80784,93840,108528,125001,143412,163932,186732,212002,239932,270732,304612,341803
%N Expansion of 1/((1-x)^4*(1-x^2)^2).
%C Equals triangle A152205 as an infinite lower triangular matrix * the triangular numbers: [1, 3, 6, ...]. - _Gary W. Adamson_, Feb 14 2010
%C a(n) is the number of partitions of n into four kinds of parts 1 and two kinds of parts 2. - _Joerg Arndt_, Mar 09 2016
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-4,10,-4,-4,4,-1).
%F a(n) = (n+4)*(2*n^4 + 32*n^3 + 172*n^2 + 352*n + 15*(-1)^n + 225)/960. - _R. J. Mathar_, Apr 01 2010
%F From _Antal Pinter_, Jan 08 2016: (Start)
%F a(n) = C(n + 3, 3) + 2*C(n + 1, 3) + 3*C(n - 1, 3) + 4*C(n - 3, 3) + ...
%F a(n) = Sum_{i = 1..z} i*C(n + 5 - 2*i,3) where z = (2*n + 3 + (-1)^n)/4.
%F (End)
%F a(n) = Sum_{i = 0..n} A002624(i). - _Antal Pinter_, May 05 2016
%t CoefficientList[Series[1/((1 - x)^4 (1 - x^2)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 14 2016 *)
%t LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {1, 4, 12, 28, 58, 108, 188, 308}, 100] (* _G. C. Greubel_, Nov 25 2016 *)
%o (PARI) Vec(1/((1-x)^4*(1-x^2)^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (Magma) [(n+4)*(2*n^4+32*n^3+172*n^2+352*n+15*(-1)^n+225)/960: n in [0..40]]; // _Vincenzo Librandi_, Feb 14 2016
%Y Cf. A152205, A001752 (for the similar series 1/((1-x)^4*(1-x^2))).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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