[seqfan] Ramanujan-Inspired Sequences for 1/pi.

Brad Klee bradklee at gmail.com
Wed Jun 17 04:43:26 CEST 2020


s=2: 1, 6, 56, 620, 7512, 96208, 1279168, 17471448, 243509720, 3447792656,
49434765888 . . .   nAn.

s=3: 1, 9, 138, 2550, 51840, 1116612, 24999408, 575368596, 13518747000,
322765065480, 7805239515216 . . .   nAn.

s=4: 1, 18, 632, 27300, 1306200, 66413424, 3515236032, 191434588488,
10648603594200, 602109586993200, 34491343330027584 . . .   nAn.

s=6: 1, 90, 20280, 5798100, 1854085464, 632693421360, 225235329359040,
82598530506097320, 30962429500615006680, 11803615010304909757680,
4560219108801622243897920 . . .   nAn.

They are all double binomial sums, but can be reduced to single sums using
Zeilberger's algorithm. Dividing a(n) by c^n for c=32,54,128, or 864,
summing
over n, we obtain either 4/pi, 9*sqrt(3)/(4*pi), 8*sqrt(2)/(3*pi) or
18/(5*pi).

These are a fairly straightforward consequence of the assertion in S.14 of:
http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf

More on this soon...

--Brad


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