A354333
a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k+1)!.
Original entry on oeis.org
1, 6, 120, 5040, 362880, 39916800, 249080832, 1307674368000, 27360571392000, 121645100408832000, 51090942171709440000, 5170403347776995328000, 15511210043330985984000000, 10888869450418352160768000000, 8841761993739701954543616000000, 432780981798838043038187520000000
Offset: 0
1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...
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Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator
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a(n) = denominator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022
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from fractions import Fraction
from math import factorial
def A354333(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022
A289488
a(n) = denominator of Sum_{k=1..n} 1/(2*k-1)!!.
Original entry on oeis.org
1, 3, 5, 105, 945, 3465, 135135, 2027025, 883575, 654729075, 13749310575, 8108567775, 718713961875, 213458046676875, 121378104973125, 191898783962510625, 372509404162520625, 73881031825566590625, 8200794532637891559375, 319830986772877770815625
Offset: 1
1, 4/3, 7/5, 148/105, 1333/945, 4888/3465, 190633/135135, 2859496/2027025, 1246447/883575, 923617228/654729075, 19395961789/13749310575, 11438644132/8108567775, ... = A289381(n)/a(n) -> A060196.
A354298
a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.
Original entry on oeis.org
1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661
Offset: 1
1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...
Cf.
A001147,
A053557,
A061354,
A064646,
A103816,
A113012,
A120265,
A143382,
A289381,
A306858,
A354299 (denominators).
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S:= 0: R:= NULL:
for n from 1 to 100 do
S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024
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Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator
Showing 1-3 of 3 results.