A334363 -id:A334363 - cp's OEIS frontend

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

1, 121, 365905, 6278929801, 358652470233121, 51516840824285500441, 15640512874253077933887601, 8915467710633236496186345872425, 8755702529258688898174686554391144001, 13878488965077362598718732163634314533105081, 33731389859841228248933904149069928786421237268881

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334363, A337725, A337726, A337727, A337729, A337730.

Table[(4 n + 1)! Sum[1/(4 k + 1)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 1)! SeriesCoefficient[(1/2) (Sin[x] + Sinh[x])/(1 - x^4), {x, 0, 4 n + 1}], {n, 0, 10}]
Table[Floor[(1/2) (Sin[1] + Sinh[1]) (4 n + 1)!], {n, 0, 10}]

a(n) = (4*n+1)!*sum(k=0, n, 1/(4*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sin(x) + sinh(x)) / (1 - x^4) = x + 121*x^5/5! + 365905*x^9/9! + 6278929801*x^13/13! + ...

a(n) = floor(c * (4*n+1)!), where c = (sin(1) + sinh(1)) / 2 = A334363.

A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

Original entry on oeis.org

5, 0, 1, 3, 8, 9, 1, 6, 4, 4, 7, 3, 5, 5, 2, 0, 3, 0, 5, 3, 8, 4, 8, 4, 5, 0, 6, 6, 5, 7, 0, 4, 2, 5, 3, 9, 4, 3, 4, 6, 0, 9, 3, 4, 5, 8, 7, 3, 9, 7, 0, 7, 3, 8, 5, 3, 3, 6, 5, 2, 4, 7, 9, 6, 6, 4, 8, 3, 4, 3, 3, 4, 1, 3, 7, 3, 7, 5, 9, 6, 7, 2, 2, 3, 1, 5, 6, 4, 3, 5, 1, 7, 4, 1, 7, 9, 9, 7, 3, 1, 2, 1, 7, 3, 7, 5, 6, 3, 7, 2

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/2! + 1/6! + 1/10! + ... = 0.5013891644735520305...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 457.

Cf. A001113, A049470, A073743, A143821, A332890, A334363, A334365.

RealDigits[(Cosh[1] - Cos[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+2)!) \\ Michel Marcus, Apr 25 2020

Equals (cosh(1) - cos(1))/2.

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Original entry on oeis.org

1, 6, 6, 8, 6, 5, 1, 0, 4, 4, 1, 7, 9, 5, 2, 4, 7, 5, 1, 1, 4, 9, 3, 9, 7, 6, 4, 4, 8, 2, 6, 5, 0, 9, 0, 7, 7, 6, 6, 5, 7, 7, 4, 6, 0, 2, 6, 7, 8, 6, 2, 4, 0, 2, 2, 7, 8, 4, 0, 6, 8, 5, 1, 5, 1, 0, 6, 9, 8, 5, 8, 1, 4, 5, 6, 5, 4, 2, 1, 1, 3, 3, 2, 9, 2, 3, 8, 8, 5, 4, 2, 7, 3, 2, 8, 5, 3, 4, 7, 9, 9, 3, 0, 3, 5, 2, 4, 7, 7, 0, 9

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/3! + 1/7! + 1/11! + ... = 0.1668651044179524751...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.
Index entries for transcendental numbers.

Cf. A001113, A049469, A073742, A332890, A334363, A334364.

RealDigits[(Sinh[1] - Sin[1])/2, 10, 111] [[1]]

suminf(k=0, 1/(4*k+3)!) \\ Michel Marcus, Apr 25 2020

Equals (sinh(1) - sin(1))/2.

A349089 a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.

Original entry on oeis.org

0, 1, 2, 6, 24, 121, 726, 5082, 40656, 365905, 3659050, 40249550, 482994600, 6278929801, 87905017214, 1318575258210, 21097204131360, 358652470233121, 6455744464196178, 122659144819727382, 2453182896394547640, 51516840824285500441, 1133370498134281009702

Offset: 0

Ilya Gutkovskiy, Mar 25 2022

nonn

Cf. A002627, A009628, A186763, A334363, A337728, A349088, A352660.

Table[n! Sum[1/(4 k + 1)!, {k, 0, Floor[(n - 1)/4]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Sin[x] + Sinh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: (sin(x) + sinh(x)) / (2*(1 - x)).

a(n) = floor(c * n!) for n > 0, where c = 1.008336089... = A334363.

cp's OEIS Frontend

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349089 a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A337728 a(n) = (4*n+1)! * Sum_{k=0..n} 1 / (4*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349089 a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.