OFFSET
0,2
COMMENTS
The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) * binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=5) = a(n).
LINKS
Project Euler, Problem 831. Triple Product
R. J. Mathar, On an alternating double sum of a triple product of aerated binomial coefficients, arXiv:2306.08022 (2023)
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).
FORMULA
G.f.: ( 1+882*x+10731*x^2-40474*x^3+36015*x^4 ) / (7*x-1)^6 .
a(n) = +42*a(n-1) -735*a(n-2) +6860*a(n-3) -36015*a(n-4) +100842*a(n-5) -117649*a(n-6).
D-finite with recurrence n*(81*n^4+360*n^3-165*n^2-640*n+404)*a(n) -7*(n+1)*(81*n^4+684*n^3+1401*n^2+434*n+40)*a(n-1)=0.
MATHEMATICA
LinearRecurrence[{42, -735, 6860, -36015, 100842, -117649}, {1, 924, 48804, 1337014, 26622288, 437049228}, 20] (* Harvey P. Dale, May 29 2023 *)
PROG
(Python)
def A361608(n): return 7**n*(n*(n*(n*(n*(81*n + 765) + 2085) + 1835) + 474) + 40)//40 # Chai Wah Wu, Mar 17 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Mar 17 2023
STATUS
approved