Search: a276593 -id:a276593
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A276592
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Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2*k-1)^(2*n).
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+10
6
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1, 1, 1, 17, 31, 691, 5461, 929569, 3202291, 221930581, 4722116521, 56963745931, 14717667114151, 2093660879252671, 86125672563201181, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 14129659550745551130667441, 16103843159579478297227731
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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a(n)/A276593(n) = (-1)^(n+1) * B_{2*n} * (2^(2*n) - 1) / (2 * (2*n)!), where B_n is the Bernoulli number. - Seiichi Manyama, Sep 03 2018
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MAPLE
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seq(numer(sum(1/(2*k-1)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..22);
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MATHEMATICA
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a[n_]:=Numerator[(-1)^n SeriesCoefficient[1/(E^x+1), {x, 0, 2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Numerator[(-1)^n Residue[Zeta[s] Gamma[s] (1-2^(1-s)), {s, 1-2 n}]] (* Steven Foster Clark, Mar 11 2023 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A276595
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Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).
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+10
4
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24, 1440, 60480, 2419200, 95800320, 2615348736000, 149448499200, 21341245685760000, 10218188434341888000, 1605715325396582400000, 28202200078783610880000, 3387648273463487338905600000, 372269041039943663616000000, 75786531374911731038945280000000
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OFFSET
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1,1
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COMMENTS
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Denominator of Bernoulli(2*n)/(2*(2*n)!). - Robert Israel, Sep 18 2016
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LINKS
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FORMULA
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MAPLE
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seq(denom(sum(1/(2*k)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..24);
seq(denom(bernoulli(2*n)/2/(2*n)!), n=1..24); # Robert Israel, Sep 18 2016
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MATHEMATICA
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Table[Denominator[Zeta[2*n]/(2*Pi)^(2*n)], {n, 1, 30}] (* Terry D. Grant, Jun 19 2018 *)
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PROG
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(PARI) a(n) = denominator(bernfrac(2*n)/(2*(2*n)!)); \\ Michel Marcus, Jul 05 2018
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A276594
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Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).
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+10
3
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1, 1, 1, 1, 1, 691, 1, 3617, 43867, 174611, 77683, 236364091, 657931, 3392780147, 1723168255201, 7709321041217, 151628697551, 26315271553053477373, 154210205991661, 261082718496449122051, 1520097643918070802691, 2530297234481911294093
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,6
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LINKS
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FORMULA
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MAPLE
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seq(numer(sum(1/(2*k)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..24);
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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