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.

A143383 Denominator of Sum_{k=0..n} 1/k!!.

Original entry on oeis.org

1, 1, 2, 6, 24, 40, 240, 560, 13440, 120960, 241920, 887040, 394240, 138378240, 276756480, 593049600, 66421555200, 4136140800, 173717913600, 14302774886400, 171633298636800, 144171970854912, 7208598542745600, 283414985441280
Offset: 0

Views

Author

Jonathan Vos Post, Aug 11 2008

Keywords

Comments

Numerators are A143382. 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) = 6 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 593049600 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)), A143382 (numerator).

Programs

  • Magma
    [n le 0 select 1 else Denominator( 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[Denominator[Sum[1/k!!, {k,0,n}]], {n,0,25}] (* G. C. Greubel, Mar 28 2019 *)
  • PARI
    vector(25, n, n--; denominator(sum(k=0,n, 1/prod(j=0,floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019
  • Sage
    [denominator(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

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

A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 7, 47, 5923, 426457, 15636757, 7318002277, 1536780478171, 603180793741, 142957467201379447, 60042136224579367741, 10127106976545720025649, 18228792557782296046168201, 12796612375563171824410077103, 3463616416319098507140327535879, 1380498543075754976417359117871773
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, A053557, A061354, A073742, A103816, A120265, A143382, A289381, A354331 (denominators), A354332, A354334.

Programs

  • Mathematica
    Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator
  • PARI
    a(n) = numerator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022
  • Python
    from fractions import Fraction
from math import factorial
def A354211(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Formula

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

A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 209594293, 1100370038249, 23023126954133, 102360822438075317, 42991545423991633141, 4350744396907953273869, 13052233190723859821607001, 9162667699888149594768114701, 7440086172309177470951709137213, 364172638960396581472899447242531
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, A053557, A061354, A103816, A120265, A143382, A354211, A354298, A354333 (denominators), A354334.

Programs

  • Mathematica
    Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator
  • PARI
    a(n) = numerator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022
  • Python
    from fractions import Fraction
from math import factorial
def A354332(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Formula

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

A354334 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.

Original entry on oeis.org

1, 3, 37, 1111, 6913, 799933, 739138093, 44841044309, 32285551902481, 9879378882159187, 1251387991740163687, 1734423756551866870183, 136771701945232930334431, 23048564587067030852654113, 42769754577382930342215977687, 409306551305554643375006906464591
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, A053557, A061354, A073743, A103816, A120265, A143382, A354211, A354332, A354335 (denominators).

Programs

  • Mathematica
    Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
Accumulate[1/(2*Range[0,20])!]//Numerator (* Harvey P. Dale, Sep 05 2024 *)
  • PARI
    a(n) = numerator(sum(k=0, n, 1/(2*k)!)); \\ Michel Marcus, May 24 2022
  • Python
    from fractions import Fraction
from math import factorial
def A354334(n): return sum(Fraction(1,factorial(2*k)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Formula

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

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

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, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).

Programs

  • Maple
    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
  • Mathematica
    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

Formula

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

A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 2, 9, 41, 1313, 5471, 1181737, 28952557, 1235309099, 150090055529, 30018011105801, 201787741322329, 523033825507476769, 44196358255381786981, 5774990812036553498851, 1949059399062336805862213, 997918412319916444601453057, 3697415655903280160125896583
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, A053557, A061354, A070910, A103816, A120265, A143382, A354303 (denominators), A354304.

Programs

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

Formula

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

A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 0, 1, 2, 43, 403, 23213, 118483, 51997111, 1842647621, 327581799289, 8918414485643, 4670006130663971, 361730891537680087, 130890931830249779173, 427294615628884602769, 6534075316966068976316143, 885163015595247156635327497, 41526561745210509140249210357
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, A053557, A061354, A073701, A091681, A103816, A120265, A143382, A354302, A354305 (denominators).

Programs

  • Mathematica
    Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Numerator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Formula

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

A354138 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k)!.

Original entry on oeis.org

1, 1, 13, 389, 4357, 1960649, 258805669, 47102631757, 11304631621681, 691843455246877, 1314502564969066301, 607300185015708631061, 335229702128671164345673, 217899306383636256824687449, 32946375125205802031892742289, 848027998784883070051677094421
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, A053557, A061354, A103816, A120265, A143382, A354211, A354332, A354334, A354378 (denominators).

Programs

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

Formula

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

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