A286286 -id:A286286 - cp's OEIS frontend

A060196 Decimal expansion of 1 + 1/(13) + 1/(135) + 1/(1357) + ...

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

Offset: 1

Evan Michael Adams (evan(AT)tampabay.rr.com), Simon Plouffe, Mar 21 2001

			1.410686134642447997690824711419115041323478...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.2, p. 423.

Harry J. Smith, Table of n, a(n) for n = 1..20000
J.-P. Allouche and T. Baruchel, Variations on an error sum function for the convergents of some powers of e, arXiv preprint arXiv:1408.2206 [math.NT], 2014.

Cf. A001147, A112293, A286286.

RealDigits[ Sqrt[E*Pi/2] * Erf[1/Sqrt[2]], 10, 107] // First
(* or *) 1/Fold[Function[2*#2-1+(-1)^#2*#2/#1], 1, Reverse[Range[100]]] // N[#, 107]& // RealDigits // First (* Jean-François Alcover, Mar 07 2013, updated Sep 19 2014 *)

{ default(realprecision, 20080); x=2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060196.txt", n, " ", d)); } \\ Harry J. Smith, Jul 02 2009

c = sqrt(e*Pi/2)*erf(1/sqrt(2)), or 2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))). - Michael Kleber, Mar 21 2001
From Peter Bala, Feb 09 2024: (Start)
Generalized continued fraction expansion:
c = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). See A286286.
c/(1 + c) = Sum_{n >= 0} (2*n-1)!!/(A112293(n) * A112293(n+1)) = 1/(1*2) + 1/(2*7) + 3/(7*36) + 15/(36*253) + 105/(253*2278) + ... = 0.5851803411..., a rapidly converging series. (End)
Equals Sum_{n >= 0} ((n - 1)*(n + 1)!*2^(n + 1))/(2*n)!. - Antonio Graciá Llorente, Feb 13 2024

More terms from Vladeta Jovovic, Mar 27 2001

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

A132424 a(0)=1 and, for n>0, a(n)=a(Floor((n-1)/a[n-1]))+2.

Original entry on oeis.org

1, 3, 3, 3, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 7, 5, 7, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 5, 7, 5, 7, 5, 7, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

John W. Layman, Aug 20 2007

nonn

Records are 1, 3, 5, 7, 9, ... and occur at {0,1,4,21,148,1333,14664,190633,2859496,48611433,...}, which appears to be A286286.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A286286.

See A130147 for a related sequence.

a[0]:=1: for n from 1 to 20000 do a[n]:=2+a[floor((n-1)/a[(n-1)])] end do: # N. J. A. Sloane, May 15 2017

A368792 a(n) = (2n-1)!! Sum_{k=0..n} k/(2*k-1)!!.

Original entry on oeis.org

0, 1, 5, 28, 200, 1805, 19861, 258200, 3873008, 65841145, 1250981765, 26270617076, 604224192760, 15105604819013, 407851330113365, 11827688573287600, 366658345771915616, 12099725410473215345, 423490389366562537093, 15669144406562813872460

Offset: 0

Seiichi Manyama, Jan 05 2024

Cf. A001147, A286286, A368793.

Table[(2n-1)!!Sum[k/(2k-1)!!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 04 2025 *)

a001147(n) = prod(k=1, n, 2*k-1);
a(n) = a001147(n)*sum(k=0, n, k/a001147(k));

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

A350557 Triangle T(n,k) read by rows with T(n,0) = (2n)! / (2^n n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2i)!) (2n)! / (2^n n!) for 0 < k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 148, 52, 10, 1, 945, 1333, 472, 96, 13, 1, 10395, 14664, 5197, 1066, 153, 16, 1, 135135, 190633, 67567, 13873, 2009, 223, 19, 1, 2027025, 2859496, 1013512, 208116, 30170, 3380, 306, 22, 1

Offset: 0

Werner Schulte, Jan 05 2022

			Triangle T(n,k) for 0 <= k <= n starts:
n\k :        0        1        2       3      4     5    6   7  8
=================================================================
  0 :        1
  1 :        1        1
  2 :        3        4        1
  3 :       15       21        7       1
  4 :      105      148       52      10      1
  5 :      945     1333      472      96     13     1
  6 :    10395    14664     5197    1066    153    16    1
  7 :   135135   190633    67567   13873   2009   223   19   1
  8 :  2027025  2859496  1013512  208116  30170  3380  306  22  1
  etc.

Cf. A001147 (column 0), A286286 (column 1), A249349 (column 2).
Cf. A000007 (alternating row sums).
Cf. A350512.

Flatten[Table[If[k==0,(2n)!/(2^n n!),Sum[Binomial[i-1,k-1]2^i i!/(2i)!,{i,k,n}](2n)!/(2^n n!)],{n,0,8},{k,0,n}]] (* Stefano Spezia, Jan 06 2022 *)

T(n,n) = 1.
T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n.
Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512.

A368794 a(n) = (2n-1)!! Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.

Original entry on oeis.org

0, 1, 2, 11, 76, 685, 7534, 97943, 1469144, 24975449, 474533530, 9965204131, 229199695012, 5729992375301, 154709794133126, 4486584029860655, 139084104925680304, 4589775462547450033, 160642141189160751154, 5943759223998947792699, 231806609735958963915260

Offset: 0

Seiichi Manyama, Jan 06 2024

Cf. A001147, A286286, A306858.

a001147(n) = prod(k=1, n, 2*k-1);
a(n) = a001147(n)*sum(k=1, n, (-1)^(k-1)/a001147(k));

a(0) = 0; a(n) = (2*n-1)*a(n-1) + (-1)^(n-1).
From Peter Bala, Feb 10 2024: (Start)
a(n) = (2*n - 2)*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)^(k-1)/(2*k-1)!! = 0.7247784590... = 1 - 1/(3 + 3/(4 + 5/(6 + 7/(8 + 9/(10 + ... ))))). (End)

cp's OEIS Frontend

A060196 Decimal expansion of 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + ...

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A112293 Row sums of number triangle A112292.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A132424 a(0)=1 and, for n>0, a(n)=a(Floor((n-1)/a[n-1]))+2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A368792 a(n) = (2*n-1)!! * Sum_{k=0..n} k/(2*k-1)!!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A368794 a(n) = (2*n-1)!! * Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A060196 Decimal expansion of 1 + 1/(13) + 1/(135) + 1/(1357) + ...

A368792 a(n) = (2n-1)!! Sum_{k=0..n} k/(2*k-1)!!.

A350557 Triangle T(n,k) read by rows with T(n,0) = (2n)! / (2^n n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2i)!) (2n)! / (2^n n!) for 0 < k <= n.

A368794 a(n) = (2n-1)!! Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.