cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A143382 Numerator of Sum_{k=0..n} 1/k!!.

Original entry on oeis.org

1, 2, 5, 17, 71, 121, 731, 1711, 41099, 370019, 740101, 2713789, 1206137, 423355111, 846710651, 1814380259, 203210595443, 12654139763, 531473870981, 43758015399281, 525096184837561, 441080795274037, 22054039763790029
Offset: 0

Views

Author

Jonathan Vos Post, Aug 11 2008

Keywords

Comments

Denominators are A143383. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:
n numerator/denominator
0 1/0!! = 1/1
1 1/0!! + 1/1!! = 2/1
2 1/0!! + 1/1!! + 1/2!! = 5/2
3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
The series converges to sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) = 3.0594074053425761445... whose decimal expansion is given by A143280. The analogs of A094007 and A094008 are determined by 2 being the only prime denominator in the convergents to the sum of reciprocals of double factorials and prime numerators beginning: a(1) = 2, a(2) = 5, a(3) = 17, a(4) = 71, a(15) = 1814380259, a(19) = 43758015399281, a(21) = 441080795274037, a(23) = 867081905243923.

Examples

			a(3) = 17 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 1814380259 because 1814380259/593049600 = 1/1 + 1/1 + 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945 + 1/3840 + 1/10395 + 1/46080 + 1/135135 + 1/645120 + 1/2027025.

Links

Crossrefs

Cf. A006882 (n!!), A094007, A143280 (m(2)), A143383 (denominators).

Programs

  • Magma
    [n le 0 select 1 else Numerator( 1 + (&+[ 1/(0 + (&*[k-2*j: j in [0..Floor((k-1)/2)]])) : k in [1..n]]) ): n in [0..25]]; // G. C. Greubel, Mar 28 2019
  • Mathematica
    Table[Numerator[Sum[1/k!!, {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Mar 28 2019 *)
Accumulate[1/Range[0,30]!!]//Numerator (* Harvey P. Dale, May 19 2023 *)
  • PARI
    vector(25, n, n--; numerator(sum(k=0,n, 1/prod(j=0,floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019
  • Sage
    [numerator(sum( 1/product((k - 2*j) for j in (0..floor((k-1)/2)))   for k in (0..n))) for n in (0..25)] # G. C. Greubel, Mar 28 2019

Formula

Numerators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).

A354331 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 6, 40, 5040, 362880, 13305600, 6227020800, 1307674368000, 513257472000, 121645100408832000, 51090942171709440000, 8617338912961658880000, 15511210043330985984000000, 10888869450418352160768000000, 2947253997913233984847872000000, 1174691236311131831103651840000000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2022

Keywords

Examples

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Crossrefs

Cf. A009445, A053556, A061355, A073742, A143383, A289488, A354211 (numerators), A354333, A354335.

Programs

  • Mathematica
    Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator
  • PARI
    a(n) = denominator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022
  • Python
    from fractions import Fraction
from math import factorial
def A354331(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Formula

Denominators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).

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

Views

Author

Ilya Gutkovskiy, May 24 2022

Keywords

Examples

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Crossrefs

Cf. A009445, A049469, A053556, A061355, A143383, A354299, A354331, A354332 (numerators), A354335.

Programs

  • Mathematica
    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
  • PARI
    a(n) = denominator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022
  • Python
    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

Formula

Denominators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).

A354335 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k)!.

Original entry on oeis.org

1, 2, 24, 720, 4480, 518400, 479001600, 29059430400, 20922789888000, 6402373705728000, 810967336058880000, 1124000727777607680000, 88635485961891348480000, 14936720782466875392000000, 27717122237428532772864000000, 265252859812191058636308480000000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2022

Keywords

Examples

			1, 3/2, 37/24, 1111/720, 6913/4480, 799933/518400, 739138093/479001600, ...

Crossrefs

Cf. A010050, A053556, A061355, A073743, A143383, A354331, A354333, A354334 (numerators).

Programs

  • Mathematica
    Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
  • PARI
    a(n) = denominator(sum(k=0, n, 1/(2*k)!)); \\ Michel Marcus, May 24 2022
  • Python
    from fractions import Fraction
from math import factorial
def A354335(n): return sum(Fraction(1,factorial(2*k)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Formula

Denominators of coefficients in expansion of cosh(sqrt(x)) / (1 - x).

A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 3, 15, 105, 189, 10395, 135135, 2027025, 34459425, 130945815, 13749310575, 316234143225, 7905853580625, 12556355686875, 1238056670725875, 776918153694375, 6332659870762850625, 7642865361265509375, 8200794532637891559375, 63966197354575554163125, 13113070457687988603440625
Offset: 1

Views

Author

Ilya Gutkovskiy, May 23 2022

Keywords

Examples

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Links

Crossrefs

Cf. A001147, A053556, A061355, A064647, A113013, A143383, A289488, A306858, A354298 (numerators).

Programs

  • Maple
    S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, denom(S);
od:
R; # Robert Israel, Jan 10 2024
  • Mathematica
    Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Denominator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Denominator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Denominator

Formula

Denominators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

A354303 a(n) is the denominator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 1, 4, 18, 576, 2400, 518400, 12700800, 541900800, 65840947200, 13168189440000, 88519495680000, 229442532802560000, 19387894021816320000, 2533351485517332480000, 855006126362099712000000, 437763136697395052544000000, 1621968544942912438272000000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 23 2022

Keywords

Examples

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Crossrefs

Cf. A001044, A006040, A053556, A061355, A070910, A143383, A354302 (numerators), A354305.

Programs

  • Mathematica
    Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Denominator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator

Formula

Denominators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

A354305 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 1, 4, 9, 192, 1800, 103680, 529200, 232243200, 8230118400, 1463132160000, 39833773056000, 20858412072960000, 1615657835151360000, 584619573580922880000, 1908495817772544000000, 29184209113159670169600000, 3953548328298349068288000000, 185476873609942457647104000000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 23 2022

Keywords

Examples

			1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...

Crossrefs

Cf. A001044, A053556, A061355, A073701, A091681, A143383, A354303, A354304 (numerators).

Programs

  • Mathematica
    Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Denominator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
Accumulate[Table[(-1)^k/(k!)^2,{k,0,20}]]//Denominator (* Harvey P. Dale, Apr 25 2023 *)

Formula

Denominators of coefficients in expansion of BesselJ(0,2*sqrt(x)) / (1 - x).

A354378 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k)!.

Original entry on oeis.org

1, 2, 24, 720, 8064, 3628800, 479001600, 87178291200, 20922789888000, 1280474741145600, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 60977668922342772100300800000, 1569543549184562477137920000000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2022

Keywords

Examples

			1, 1/2, 13/24, 389/720, 4357/8064, 1960649/3628800, 258805669/479001600, ...

Crossrefs

Cf. A010050, A049470, A053556, A061355, A143383, A354138 (numerators), A354331, A354333, A354335.

Programs

  • Mathematica
    Table[Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Cos[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
  • PARI
    a(n) = denominator(sum(k=0, n, (-1)^k/(2*k)!)); \\ Michel Marcus, May 24 2022

Formula

Denominators of coefficients in expansion of cos(sqrt(x)) / (1 - x).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.