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%I #94 Apr 29 2017 12:26:59
%S 0,5,80,405,1280,3125,6480,12005,20480,32805,50000,73205,103680,
%T 142805,192080,253125,327680,417605,524880,651605,800000,972405,
%U 1171280,1399205,1658880,1953125,2284880,2657205,3073280,3536405,4050000,4617605,5242880,5929605
%N a(n) = 5*n^4.
%C More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).
%C More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.
%H Ilya Gutkovskiy, <a href="/A269792/a269792_3.pdf">Examples of the ordinary generating function for the values of quartic polynomial</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)
%F G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
%F E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
%F a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F a(n) = 5*A000583(n) = A008587(n)*A000578(n).
%F Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...
%p A269792:=n->5*n^4: seq(A269792(n), n=0..50); # _Wesley Ivan Hurt_, Apr 28 2017
%t Table[5 n^4, {n, 0, 33}]
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]
%o (PARI) x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ _Altug Alkan_, Mar 31 2016
%Y Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Mar 31 2016
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