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

1, 30, 181, 600, 1501, 3150, 5881, 10080, 16201, 24750, 36301, 51480, 70981, 95550, 126001, 163200, 208081, 261630, 324901, 399000, 485101, 584430, 698281, 828000, 975001, 1140750, 1326781, 1534680, 1766101, 2022750, 2306401, 2618880, 2962081, 3337950, 3748501, 4195800

Offset: 1

Author

Stefano Maruelli, Feb 05 2017

Keywords

• nonn
• easy

Comments

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.

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.

Examples

			For n=2, a(2) = (2^5 - 1^5) - (1) = 30.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
• Stefano Maruelli, Trapezoidal gnomon roof, case n=5.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Programs

Mathematica

LinearRecurrence[{4,-5,0,5,-4,1},{1,30,181,600,1501,3150},40] (* Harvey P. Dale, May 03 2024 *)

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

Formula

G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/((1 + x)*(1 - x)^5).

a(n) = (5*(n^2 - 1)*n^2 - (-1)^n + 1)/2.

a(n) = (n^5-(n-1)^5) - a(n-1).

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