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Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
1

%I #40 Nov 19 2017 02:42:15

%S 0,1,4,9,21,35,65,95,155,210,315,406,574,714,966,1170,1530,1815,2310,

%T 2695,3355,3861,4719,5369,6461,7280,8645,9660,11340,12580,14620,16116,

%U 18564,20349,23256,25365,28785,31255,35245,38115,42735,46046,51359,55154,61226,65550,72450,77350,85150,90675,99450,105651,115479

%N Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

%F G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

%F a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).

%F Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...

%F Convolution of the sequences A027656 and A085787.

%F a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - _Luce ETIENNE_, Nov 18 2017

%t CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]

%t LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

%Y Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Aug 15 2017