# [seqfan] Re: A series for exp(Pi)

Alexander Povolotsky apovolot at gmail.com
Sat Oct 24 20:59:31 CEST 2009

```It appears that conjectured closed form expression obtained in the
previous email could be further simplified to

a(n)=
(3*2^(-3 + n)*gamma(((-1 - 6*i) + n)/2)*gamma(((-1 + 6*i) + n)/2)*((1
+ (-1)^n)*csch(3*pi) - (-1 + (-1)^n)*sech(3*pi))*sinh(6*pi))/pi

Alexander R. Povolotsky

On 10/24/09, Alexander Povolotsky <apovolot at gmail.com> wrote:
> WolframAlpha gives
> Possible closed form for 19 terms given:
> a_n = (2^(n-2)*(3*(-1)^n*
> Gamma(-3*i)*Gamma(3*i)-(-1)^n*Gamma(1/2-3*i)*Gamma(1/2+3*i)+3*Gamma(-3*i)*Gamma(3*i)+Gamma(1/2-3*i)*Gamma(1/2+3*i))*Gamma(n/2-(1/2+3*i))*Gamma(n/2-(1/2-3*i)))/(Gamma(-3*i)*Gamma(3*i)*
> Gamma(1/2-3*i)* Gamma(1/2+3*i))
>
> On 10/24/09, Simon Plouffe <simon.plouffe at gmail.com> wrote:
>> =====================================================
>>
>> Dear Seqfans,
>>
>> http://research.att.com/~njas/sequences/A166748
>> has a(0)=1, a(1)=6 and a(n)=(40-4*n+n2)*a(n-2) for n>=2.
>> ( See http://research.att.com/~njas/sequences/A039661 )
>>
>> Maybe I should already know, but... how is the closed form of this
>> kind of recurrences found?
>> ==================================================================
>>
>>
>>
>> Hello
>>
>>   the sequence is easily cracked by gfun,
>>
>> [1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000,
>> 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750,
>>
>>
>>      3061947432960000, 50908949029311750, 894088650424320000,
>> 16545408434526318750, 321871914152755200000, 6568527148506948543750,
>>
>>      140336154570601267200000, 3133187449837814455368750,
>> 72974800376712658944000000, 1770250909158365167283343750,
>>
>>      44660577830548147273728000000, 1170135850953679375574290218750,
>> 31798331415350280858894336000000, 895153925979564722314332017343750,
>>
>>      26074631760587230304293355520000000,
>> 785049993084078261469669179210468750,
>> 24405855327909647564818580766720000000,
>>
>>      782694843104826026685260171672837343750,
>> 25870206647584226418707695612723200000000,
>> 880531698492929280020917693131942011718750,
>>
>>      30837286323920397891099573170366054400000000,
>> 1110350471799583822106377211039378876777343750,
>>
>>      41075265383461969990944631462927584460800000000,
>> 1560042412878415270059459981510327321872167968750,
>>
>>      60791392767523715586598054565132825001984000000000,
>> 2428986036851692575482579191211579640154965527343750,
>>
>>      99454718567668798699674417268557301703245824000000000,
>> 4170569025274356152103588471310282242146075810449218750]
>>
>>  > listtorec(%,a(n));
>>                      2      3                          2    3
>> [{(216 + 216 n + 6 n  + 6 n ) a(n) + (111 + 43 n + 5 n  + n ) a(n + 1) +
>> (-6 - 6 n) a(n + 2) + (-n - 3) a(n + 3), 1 = 6, 6 = 36, a(0) = 1}, ogf
>>
>>      ]
>>
>> simon plouffe
>>
>>
>>
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>>
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>>
>

```