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%I #20 Sep 08 2022 08:46:15
%S 1,2,3,4,20,45,109,275,708,1765,4442,11196,28207,70985,178755,450130,
%T 1133423,2853888,7186144,18094709,45562353,114725755,288879164,
%U 727396569,1831581574,4611915224,11612784735,29240946181,73628587619,185396495082
%N Expansion of (1 + x - x^2 - 6*x^3)/(1 - x - 2*x^2 - 3*x^3 - 4*x^4).
%C In general, the ordinary generating function for the recurrence relation b(n) = b(n - 1) + 2*b(n - 2) + 3*b(n - 3) + 4*b(n - 4) + ... + k*b(n - k), with n > k - 1 and initial values b(i-1) = i for i = 1..k, is (Sum_{m = 0..(k - 1)} (-m^3 - 3*m^2 + 4*m + 6)*x^m/6)/(1 - Sum_{m = 1..k} m*x^m).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,3,4).
%F G.f.: (1 + x - x^2 - 6*x^3)/(1 - x - 2*x^2 - 3*x^3 - 4*x^4).
%t LinearRecurrence[{1, 2, 3, 4}, {1, 2, 3, 4}, 30]
%t CoefficientList[Series[(1 + x - x^2 - 6 x^3) / (1 - x - 2 x^2 - 3 x^3 - 4 x^4), {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 04 2016 *)
%o (PARI) Vec((1+x-x^2-6*x^3)/(1-x-2*x^2-3*x^3-4*x^4) + O(x^40)) \\ _Michel Marcus_, Feb 02 2016
%o (Magma) [n le 4 select n else Self(n-1)+2*Self(n-2)+3*Self(n-3)+4*Self(n-4): n in [1..35]]; // _Vincenzo Librandi_, Feb 04 2016
%Y Cf. A115451, A111586.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Feb 02 2016
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