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%I #96 Mar 29 2018 09:54:37
%S 0,1,5,10,26,40,80,110,190,245,385,476,700,840,1176,1380,1860,2145,
%T 2805,3190,4070,4576,5720,6370,7826,8645,10465,11480,13720,14960,
%U 17680,19176,22440,24225,28101,30210,34770,37240,42560,45430,51590,54901,61985,65780,73876,78200,87400,92300,102700,108225,119925,126126
%N Expansion of x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).
%C More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).
%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 + 4*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 (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...
%F Convolution of the sequences A027656 and A001082 (with offset 0).
%F a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - _Luce ETIENNE_, Nov 18 2017
%t CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
%t LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ _Altug Alkan_, Aug 15 2017
%Y Cf. A001082, A002414, A002419, A002624, A027656, A195152, A287143.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Aug 15 2017
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