cp's OEIS Frontend

A281999 Half of the height of the right trapezoidal gnomon (of the derivative of Y=X^5).

%I A281999 #57 May 03 2024 19:17:25
%S A281999 1,30,181,600,1501,3150,5881,10080,16201,24750,36301,51480,70981,
%T A281999 95550,126001,163200,208081,261630,324901,399000,485101,584430,698281,
%U A281999 828000,975001,1140750,1326781,1534680,1766101,2022750,2306401,2618880,2962081,3337950,3748501,4195800
%N A281999 Half of the height of the right trapezoidal gnomon (of the derivative of Y=X^5).
%C A281999 The curves Y = X^m are characterized by the fact that the first derivative Y'= m*X^(m-1) (and all the following derivatives) are squarable in the integers by rectangular columns called gnomons with base=1 and height M_m = X^m - (X-1)^m. Calling Y' = X^m - (X-1)^m the first "integer" derivative, considering the case m=5, {a(n)} represents the values of half of the maximum (right) height of the trapezoidal gnomons. The formula is: a(n) = (n^5 - (n-1)^5) - a(n-1). The broken line given by joining the points (n; 2*a(n)); define a series of trapezoidal areas (gnomons) that have the same area below the curve Y'=5*X^4. It means that the recursive sum of the trapezoidal gnomon's area, (a(n) + a(n-1))*1, from 1 to n, gives n^5.
%C A281999 The general formula, changing the exponent for all the Y = X^m curves, gives infinitely many new sequences: b(m,k) = m^k - (m-1)^k - b(m-1,k). The same can be done for all the following derivatives. For the smallest exponents k of Y = X^k the sequences are known: for k=3 the sequence is A032528, for k=4 the sequence is A007588, and k=5 corresponds to this sequence.
%H A281999 Colin Barker, <a href="/A281999/b281999.txt">Table of n, a(n) for n = 1..1000</a>
%H A281999 Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, <a href="https://arxiv.org/abs/2301.09765">Enumeration of multi-rooted plane trees</a>, arXiv:2301.09765 [math.CO], 2023.
%H A281999 Stefano Maruelli, <a href="https://web.archive.org/web/20171120113303/http://maruelli.com/two-hand-clock/MARUELLI-TRAPEZOIDAL-GNOMON-ROOF-INTEGER-VALUE-N5.jpg">Trapezoidal gnomon roof, case n=5</a>.
%H A281999 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A281999 G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/((1 + x)*(1 - x)^5).
%F A281999 a(n) = (5*(n^2 - 1)*n^2 - (-1)^n + 1)/2.
%F A281999 a(n) = (n^5-(n-1)^5) - a(n-1).
%F A281999 a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n>6. - _Colin Barker_, Feb 27 2017
%e A281999 For n=2, a(2) = (2^5 - 1^5) - (1) = 30.
%t A281999 LinearRecurrence[{4,-5,0,5,-4,1},{1,30,181,600,1501,3150},40] (* _Harvey P. Dale_, May 03 2024 *)
%o A281999 (PARI) Vec(x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/((1 + x)*(1 - x)^5) + O(x^30)) \\ _Colin Barker_, Feb 27 2017
%K A281999 nonn,easy
%O A281999 1,2
%A A281999 _Stefano Maruelli_, Feb 05 2017