cp's OEIS Frontend

A331429 Expansion of x^2*(10-5*x+x^2)/((1-x)^4*(1-x^2)).

%I A331429 #33 Aug 17 2025 01:25:19
%S A331429 0,0,10,35,91,189,351,594,946,1430,2080,2925,4005,5355,7021,9044,
%T A331429 11476,14364,17766,21735,26335,31625,37675,44550,52326,61074,70876,
%U A331429 81809,93961,107415,122265,138600,156520,176120,197506,220779,246051,273429,303031,334970,369370,406350,446040,488565,534061,582659,634501
%N A331429 Expansion of x^2*(10-5*x+x^2)/((1-x)^4*(1-x^2)).
%C A331429 Column 2 of triangle in A331432.
%D A331429 J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
%H A331429 G. C. Greubel, <a href="/A331429/b331429.txt">Table of n, a(n) for n = 0..1000</a>
%H A331429 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A331429 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%H A331429 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A331429 a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n>5. - _Vincenzo Librandi_, Jan 17 2020
%F A331429 From _Bruno Berselli_, Jan 17 2020: (Start)
%F A331429 a(n) = (n*(n + 3)*(n^2 + 3*n - 2) + 4*(-1)^n - 4)/8. Therefore:
%F A331429 a(n) = n*(n + 3)*(n^2 + 3*n - 2)/8 if n is even,
%F A331429 a(n) = n*(n + 3)*(n^2 + 3*n - 2)/8 - 1 if n is odd. (End)
%F A331429 E.g.f.: (1/8)*(4*exp(-x) + (-4 + 8*x + 32*x^2 + 12*x^3 + x^4)*exp(x)). - _G. C. Greubel_, Mar 22 2022
%t A331429 CoefficientList[Series[x^2(10-5x+x^2)/((1-x)^4(1-x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jan 17 2020 *)
%t A331429 Table[(n(n+3)(n^2+3n-2) +4(-1)^n -4)/8, {n, 0, 50}] (* _Bruno Berselli_, Jan 17 2020 *)
%o A331429 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0] cat Coefficients(R!( x^2*(10-5*x+x^2)/((1-x)^4*(1-x^2)))); // _Vincenzo Librandi_, Jan 17 2020
%o A331429 (Sage) [n*(n+3)*(n^2 +3*n -2)/8 - (n%2) for n in (0..50)] # _G. C. Greubel_, Mar 22 2022
%Y A331429 Cf. A331432.
%K A331429 nonn,easy
%O A331429 0,3
%A A331429 _N. J. A. Sloane_, Jan 16 2020