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opelly27
40 lines
1.5 KiB
INI
40 lines
1.5 KiB
INI
// y-cruncher Custom Formula File
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//
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// This can be loaded directly from the Custom Compute menu or
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// entered from the command line as "custom:filename".
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//
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// y-cruncher custom custom:"constant - algorithm"
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//
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//
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// Author: Alexander J. Yee
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// Date: September 29, 2018
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//
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// Value: 0.91596559417721901505460351493238411077414937428167...
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//
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// Formula by Pilehrood (2010):
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//
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// 1 inf ( (-256)^k (419840k ^6 - 915456k^5 + 782848k^4 - 332800k^3 + 73256k^2 - 7800k + 315) k!^2 (4k)!^4 )
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// Catalan = ---- SUM (------------------------------------------------------------------------------------------------)
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// 64 k=0 ( k^3 (2k-1) (4k-1)^2 (4k-3)^2 (8k)!^2 (2k)! )
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//
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// https://hal.inria.fr/hal-00990465/document
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// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#catalan_pilehrood_long
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//
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{
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NameShort : "Catalan"
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NameLong : "Catalan's Constant"
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AlgorithmShort : "Pilehrood (long)"
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AlgorithmLong : "Pilehrood (2010) (long)"
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Formula : {
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SeriesHypergeometric : {
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CoefficientP : -1
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CoefficientQ : 0
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CoefficientD : 2
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PolynomialP : [315 -7800 73256 -332800 782848 -915456 419840]
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PolynomialQ : [-11025 295680 -3138304 17219584 -54222848 101711872 -112197632 67108864 -16777216]
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PolynomialR : [0 0 0 -288 3648 -17408 38912 -40960 16384]
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}
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}
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}
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