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

[Alexander Povolotsky]

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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>