OFFSET
1,2
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.
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).
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
EXAMPLE
For n=2, a(2) = (2^5 - 1^5) - (1) = 30.
MATHEMATICA
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {1, 30, 181, 600, 1501, 3150}, 40] (* Harvey P. Dale, May 03 2024 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefano Maruelli, Feb 05 2017
STATUS
approved