# Finding a sequence function

I would appreciate your help in finding a polynomial sequence function based on a set of data points. For each pair of integers {j, k}, I have the following data points

data = {
{{0, 1}, (b^2/(36 m))},
{{1, 1}, (b^2)/(900 m^3)},
{{2, 1}, (b^2)/(2940 m^5)},
{{3, 1}, (b^2)/(3780 m^7)},
{{4, 1}, (5 b^2)/(13068 m^9)},
{{5, 1}, (691 b^2)/(780780 m^11)},
{{0, 2}, (b^4/(600 m^5)},
{{1, 2}, (53 b^4)/(26460 m^7)},
{{2, 2}, (b^4 )/(264 m^9)},
{{3, 2}, (1769 b^4)/(165165 m^11)},
{{4, 2}, (12959 b^4)/(304200 m^13)}}


I am seeking a polynomial function that depends on Bernoulli numbers and factorials of j and k. Specifically, b is raised to the power of k such as (b)^(2k), and m is raised to the power of j and k, such as (m^2)^(-2k-j). The entries provided are just a subset, and ideally, the function should account for an infinite sum over j and k.

I attempted to use FindSequenceFunction in Mathematica, but it could not help. Is there any other way to try in Mathemtica?

• The closing parenthesis is missing, {0, 2}, (b^4/(600 m^5) -> {0, 2}, (b^4)/(600 m^5) Commented Jul 16 at 10:31
• Indeed, that is true!
– Hawi
Commented Jul 16 at 10:55

I found the formula for the first part of your data

 data1 = {{{0, 1}, (b^2/(36  m))}, {{1, 1}, (b^2)/(900  m^3)},
{{2, 1}, (b^2)/(2940  m^5)}, {{3, 1}, (b^2)/(3780  m^7)},
{{4, 1}, (5  b^2)/(13068  m^9)}, {{5, 1}, (691  b^2)/(780780  m^11)}};


The coefficients are

 Table[data1[[k, 2]]/b^2*m^(2 k - 1), {k, 1, 6}]
(* {1/36, 1/900, 1/2940, 1/3780, 5/13068, 691/780780} *)


and from there (k from 1):

 Table[data1[[k, 2]]/b^2* m^(2 k - 1) /
((-1)^(k + 1)*BernoulliB[2 k]/(2*(4*k^2 - 1))), {k, 1, 6}]

(* {1, 1, 1, 1, 1, 1} *)

• Thanks a lot, I actually had that part already but forgot to mention it. Great job! This could be generalized to {k,1,Infinity} and it correctly reproduces the coeffs.
– Hawi
Commented Jul 16 at 10:53