Timeline for this sequence $A_{n}$ have recursive relations?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Dec 20, 2014 at 16:49 | comment | added | Michael Stoll | @math110: I should perhaps clarify that I have not actually proved the recurrence, I have only checked it for $n$ up to 490 or so. This is certainly very strong evidence that it is true, but does not constitute a proof. | |
| Dec 20, 2014 at 16:37 | comment | added | math110 | @MichaelStoll,Thank you for give this usefull reslut | |
| Dec 20, 2014 at 15:44 | comment | added | Michael Stoll | The (likely) recurrence above suggests that considering $b_n = (n+2) a_n = (n+2) (2n+3)! A_{n+3}$ might simplify the computation (the degree of the coefficients in the recurrence goes down from 5 to 4). | |
| Dec 20, 2014 at 15:38 | comment | added | Michael Stoll | A small correction: the coefficient of $a_{n+5}$ has a factor $(n+7)^2$ instead of $n+7$. | |
| Dec 20, 2014 at 13:19 | comment | added | fedja | +1 for bringing people's attention to that brilliant (though, alas, so far totally useless for my favorite kind of "formal algebra" questions) text again. | |
| Dec 20, 2014 at 12:43 | comment | added | Michael Stoll | See the book "A=B" by Petkovsek, Wilf and Zeilberger for background and ways of deriving such recurrences in concrete cases: math.upenn.edu/~wilf/AeqB.html. | |
| Dec 20, 2014 at 12:40 | comment | added | Michael Stoll | Then experimentally, $2^4 (n+2) (n+3) (2n+3) (2n+7) (27296942n+96266479) a_n +$ $2^3 (n+3) (1373740004n^4 + 11974782476n^3 + 36773573779n^2 + 46446616891n +$ $19095420600) a_{n+1} - 2^2 (n+4) (1258873802n^4 + 12734573713n^3 +$ $46301208703n^2 + 68411873198n + 30162171666) a_{n+2} + 2 (n+5) (1272271211n^4 + 21664023114n^3 + 138039728785n^2 + 392255730894n +$ $423165299544) a_{n+3} - (n+6) (169792631n^4 + 623043934n^3 - 19045733677n^2 -$ $151948512276n - 311659470468) a_{n+4} -$ $3 (n+7) (3n+19) (3n+20) (3057095n + 14424538) a_{n+5} = 0$ for all $n \ge 0$. | |
| Dec 20, 2014 at 12:36 | comment | added | Michael Stoll | By general theory, your sequence has to satisfy a recurrence of the desired form where the coefficients are polynomials in $n$. Let $a_n=(2n+3)!A_{n+3}$. See next comment... | |
| Dec 20, 2014 at 9:43 | history | asked | math110 | CC BY-SA 3.0 |