A060196 -id:A060196 - cp's OEIS frontend

A286286 a(0) = 0; thereafter, a(n) = (2n-1)a(n-1) + 1.

0, 1, 4, 21, 148, 1333, 14664, 190633, 2859496, 48611433, 923617228, 19395961789, 446107121148, 11152678028701, 301122306774928, 8732546896472913, 270708953790660304, 8933395475091790033, 312668841628212651156, 11568747140243868092773

Offset: 0

N. J. A. Sloane, May 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..404

Conjectured to give indices of records in A132424.

Cf. A001147, A002627 (similar sequence), A000522, A060196.

NestList[{(2 #2 - 1) #1 + 1, #2 + 1} & @@ # &, {0, 1}, 19][[All, 1]] (* Michael De Vlieger, Dec 10 2021 *)

a(n) = (2*n-1)!! * Sum_{k=1..n} 1/(2*k-1)!!. - Seiichi Manyama, Sep 02 2017
a(n) = floor((2*n-1)!!*A060196), for n > 0. - Peter McNair, Dec 10 2021
From Peter Bala, Feb 09 2024: (Start)
a(n) = 2*n*a(n-1) - (2*n - 3)*a(n-2) with a(0) = 0 and a(1) = 1.
The double factorial numbers (2*n-1)!! = A001147(n) satisfy the same recurrence, leading to the generalized continued fraction expansion Limit_{n -> oo} a(n)/(2*n-1)!! = Sum_{k >= 1} 1/(2*k-1)!! = A060196 = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). (End)

A108088 Decimal expansion of 1/(1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))).

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Jun 21 2005

Term of Ramanujan's formula (see A059444 and A060196).

			0.6556795424187984715438712307308112833992823328704...

S. R. Finch, "Mathematical Constants", Cambridge, pp. 423-428.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A111129.

RealDigits[Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]], 10, 111][[1]]

sqrt(Pi*exp(1)/2)*erfc(1/sqrt(2)) \\ G. C. Greubel, Feb 03 2017

Equals sqrt(Pi*e/2)*erfc(1/sqrt(2)), where erfc is the complementary error function. - Daniel Forgues, Apr 14 2011

Also equals Integral_{-infinity..infinity} (1/sqrt(2*Pi))*exp(-x^2/2)/(1+x^2) dx, where the integrand is normal PDF times Cauchy PDF. - Jean-François Alcover, Apr 28 2015

A112293 Row sums of number triangle A112292.

Original entry on oeis.org

1, 2, 7, 36, 253, 2278, 25059, 325768, 4886521, 83070858, 1578346303, 33145272364, 762341264373, 19058531609326, 514580353451803, 14922830250102288, 462607737753170929, 15266055345854640658, 534311937104912423031

Offset: 0

Paul Barry, Sep 01 2005

Seiichi Manyama, Table of n, a(n) for n = 0..404

Cf. A001147, A060196, A286286.

T[n_, k_] := If[k <= n, (2 n - 1)!! / (2 k - 1)!!, 0];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jun 13 2019 *)

a(n) = Sum_{k=0..n} (2n-1)!!/(2k-1)!!.
a(n) = Sum_{k=0..n} 2^(n-k)(n-1/2)!/(k-1/2)!.
a(n) = Sum_{k=0..n} n!C(2n, n)2^(k-n)/(k!C(2k, k)).
a(n) = Sum_{k=0..n} (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)).
Conjecture: a(n) - 2*n*a(n-1) + (2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 28 2014
a(n) = floor((2*n-1)!! * C) for n>0, where C = 1 + sqrt(e*Pi/2)*erf(1/sqrt(2)). - Don Knuth, Mar 25 2018
a(n) = 2^n*(C*Gamma(n + 1/2) + Gamma(n + 1/2, 1/2))*sqrt(e/2) for n >= 0, where C = sqrt(2/(e*Pi)) - erfc(1/sqrt(2)). - Peter Luschny, Mar 25 2018
a(n) = a(n-1) * (2*n-1) + 1 for n > 0 and a(0) = 1; that proves the conjecture of R. J. Mathar from Nov 28 2014; G.f. A(x) satisfies the equation: A(x) = 1/(1-x)^2 + A'(x) * 2*x^2/(1-x), where A' is the first derivative of A. - Werner Schulte, Oct 18 2023

A059444 Decimal expansion of square root of (Pi * e / 2).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 01 2001

Appears as constant factor in Proposition 1.12, p. 5, of Feige et al. (2007). - Jonathan Vos Post, Jun 18 2007

			2.066365677...

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, Oxford and NY, 2001, page 68.

Harry J. Smith, Table of n, a(n) for n = 1..20000
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Uri Feige, Guy Kindler, Ryan O Donnell, Understanding Parallel Repetition Requires Understanding Foams, Electronic Colloquium on Computational Complexity, Report TR07-043 (ISSN 1433-8092, 14th Year, 43rd Report), 7 May 2007.
OEIS Wiki, A remarkable formula of Ramanujan

Cf. A059445, A060196, A108088.

RealDigits[N[Sqrt[ \[Pi]*\[ExponentialE]/2], 100]][[1]]
RealDigits[Sqrt[(Pi*E)/2],10,120][[1]] (* Harvey P. Dale, Nov 27 2024 *)

{ default(realprecision, 20080); x=sqrt(Pi*exp(1)/2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b059444.txt", n, " ", d)); } \\ Harry J. Smith, Jun 27 2009

Sqrt(Pi*e/2) = A + B with A = 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + 1/(1*3*5*7*9) + ... = 1.410686134... (see A060196) and B = 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4/(1 + 5/(1 + ...)))))) = 0.65567954241... (see A108088) - (S. Ramanujan)

Equals (sqrt(2)*exp(1/4)*(sum(n>=0, n!/(2*n)! ) - 1))/erf(1/2). - Jean-François Alcover, Mar 22 2013

Edited by Daniel Forgues, Apr 14 2011

A289381 a(n) = numerator of Sum_{k=1..n} 1/(2*k-1)!!.

Original entry on oeis.org

1, 4, 7, 148, 1333, 4888, 190633, 2859496, 1246447, 923617228, 19395961789, 11438644132, 1013879820791, 301122306774928, 171226409734763, 270708953790660304, 525493851475987649, 104222947209404217052, 11568747140243868092773, 451181138469510855618148

Offset: 1

Seiichi Manyama, Sep 02 2017

			1, 4/3, 7/5, 148/105, 1333/945, 4888/3465, 190633/135135, 2859496/2027025, 1246447/883575, 923617228/654729075, 19395961789/13749310575, 11438644132/8108567775, ... = a(n)/A289488(n) -> A060196.

Seiichi Manyama, Table of n, a(n) for n = 1..404
OEIS Wiki, A remarkable formula of Ramanujan

Cf. A001147, A060196, A289488, A354298.

Numerators of coefficients in expansion of sqrt(Pi*x*exp(x)/2) * erf(sqrt(x/2)) / (1 - x). - Ilya Gutkovskiy, 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

Seiichi Manyama, Sep 02 2017

			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.

Seiichi Manyama, Table of n, a(n) for n = 1..404
OEIS Wiki, A remarkable formula of Ramanujan

Cf. A001147, A060196, A289381, A354299.

Accumulate[Table[1/(2k-1)!!,{k,20}]]//Denominator (* Harvey P. Dale, Mar 01 2023 *)

Denominators of coefficients in expansion of sqrt(Pi*x*exp(x)/2) * erf(sqrt(x/2)) / (1 - x). - Ilya Gutkovskiy, May 24 2022

A306858 Decimal expansion of 1 - 1/(13) + 1/(135) - 1/(1357) + ...

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

			0.7247784590070763318182279676062161663121329...

Cf. A001147, A014481, A060196, A068996, A092605, A113014.

RealDigits[Sqrt[Pi/(2 Exp[1])] Erfi[1/Sqrt[2]], 10, 110] [[1]]
RealDigits[Sqrt[2] DawsonF[1/Sqrt[2]], 10, 110] [[1]]

Equals sqrt(Pi/(2*exp(1)))*erfi(1/sqrt(2)), where erfi is the imaginary error function.
Equals (1/sqrt(e)) * Sum_{k>=0} 1/(2^k * k! * (2*k+1)) = 1/(sqrt(e)) * Sum_{k>=0} 1/A014481(k). - Amiram Eldar, Nov 12 2021
Equals 1/(1+A113014). - Jon Maiga, Nov 12 2021

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 16 2013

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015
Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Engel Expansion
Googology Wiki, Double factorial

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019

cp's OEIS Frontend

A286286 a(0) = 0; thereafter, a(n) = (2*n-1)*a(n-1) + 1.

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A108088 Decimal expansion of 1/(1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))).

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A112293 Row sums of number triangle A112292.

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A059444 Decimal expansion of square root of (Pi * e / 2).

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A289381 a(n) = numerator of Sum_{k=1..n} 1/(2*k-1)!!.

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A289488 a(n) = denominator of Sum_{k=1..n} 1/(2*k-1)!!.

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A306858 Decimal expansion of 1 - 1/(1*3) + 1/(1*3*5) - 1/(1*3*5*7) + ...

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A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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A286286 a(0) = 0; thereafter, a(n) = (2n-1)a(n-1) + 1.

A306858 Decimal expansion of 1 - 1/(13) + 1/(135) - 1/(1357) + ...