A178647 -id:A178647 - cp's OEIS frontend

A051577 a(n) = (2*n + 3)!!/3 = A001147(n+2)/3.

1, 5, 35, 315, 3465, 45045, 675675, 11486475, 218243025, 4583103525, 105411381075, 2635284526875, 71152682225625, 2063427784543125, 63966261320836875, 2110886623587616875, 73881031825566590625, 2733598177545963853125, 106610328924292590271875

Offset: 0

Wolfdieter Lang

Row m = 3 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..250
H. W. Gould, Harris Kwong, and Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), Article 15.9.6.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 10.
Eric Weisstein's World of Mathematics, Double Factorial.
Wikipedia, Double factorial.

Cf. A000165, A001147, A002866(n+1) (m=0..2 rows of A(3; m,n)).

Cf. A051578, A051579, A051580, A051581, A051582, A051583, A178647.

F:=Factorial;; List([0..25], n-> F(2*n+4)/(12*2^n*F(n+2)) ); # G. C. Greubel, Nov 12 2019

F:=Factorial; [F(2*n+4)/(12*2^n*F(n+2)): n in [0..25]]; // G. C. Greubel, Nov 12 2019

seq( doublefactorial(2*n+3)/3,n=0..10) ; # R. J. Mathar, Sep 29 2013

Table[(2*n + 3)!!/3!!, {n, 0, 25}] (* G. C. Greubel, Jan 22 2017 *)
a[n_] := Sum[(-1)^k*Binomial[2*n + 1, n + k]*StirlingS1[n + k + 1 ,k], {k , 1, n + 1}]; Flatten[Table[a[n], {n, 0, 18}]] (* Detlef Meya, Jan 17 2024 *)

vector(26, n, (2*n+2)!/(6*2^n*(n+1)!) ) \\ G. C. Greubel, Nov 12 2019

f=factorial; [f(2*n+4)/(12*2^n*f(n+2)) for n in (0..25)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n + 3)!!/3!!.
E.g.f.: 1/(1 - 2*x)^(5/2).
a(n) ~ (4/3) * sqrt(2) * n^2 * 2^n * e^(-n) * n^n *{1 + (47/24)*n^(-1) + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
Ramanujan polynomials -psi_n(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004
a(n) = 2^(2 + n) * Gamma(n + 5/2)/(3 * sqrt(Pi)). - Gerson Washiski Barbosa, May 05 2010
From Peter Bala, May 26 2017: (Start)
D-finite with recurrence a(n+1) = (2*n + 5)*a(n) with a(0) = 1.
O.g.f. A(x) satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 5*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 5*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - ... - (2*n+3)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes, 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 5*x/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 3*(sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1), where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 3*(1 - sqrt(Pi/(2*e)) * erfi(1/sqrt(2))), where erfi is the imaginary error function. (End)
a(n) = Sum_{k=1..n+1} (-1)^k*binomial(2*n + 1, n + k)*Stirling1(n + k + 1, k). - Detlef Meya, Jan 17 2024

A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 7, 63, 693, 9009, 135135, 2297295, 43648605, 916620705, 21082276215, 527056905375, 14230536445125, 412685556908625, 12793252264167375, 422177324717523375, 14776206365113318125, 546719635509192770625

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578 (rows m=0..4).

Cf. A051580, A051581, A051582, A051583, A178647.

List([0..20], n-> Product([0..n-1], j-> 2*j+7) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+7: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+5)/df(5), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[7/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+5) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+7) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+5)!!/4!!.
E.g.f.: 1/(1-2*x)^(7/2).
a(n) ~ 8/15*sqrt(2)*n^3*2^n*e^-n*n^n*(1 + 107/24*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: G(0)/(10*x) -1/(5*x), where G(k)= 1 + 1/(1 - x*(2*k+5)/(x*(2*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 7)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 7*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 7*x/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 15 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 20, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 15 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 10, where erfi is the imaginary error function. (End)

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 11, 143, 2145, 36465, 692835, 14549535, 334639305, 8365982625, 225881530875, 6550564395375, 203067496256625, 6701227376468625, 234542958176401875, 8678089452526869375, 338445488648547905625, 13876265034590464130625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..398
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A178647.

Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).

List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(11/2,n), n = 0..20); # G. C. Greubel, Nov 12 2019

(2*Range[0,20]+9)!!/945 (* Harvey P. Dale, Apr 10 2019 *)
Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+9)!!/9!!.
E.g.f.: 1/(1-2*x)^(11/2).
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)

A110894 Decimal expansion of error function of square root of 2.

Original entry on oeis.org

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

Offset: 0

Joost de Winter, Sep 20 2005

			0.9544997...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Wikipedia, 68-95-99.7 rule
Wikipedia, Normal distribution
Wikipedia, Standard deviation

Cf. A178647 (1sigma), A270712 (3sigma).

erf(sqrt(2.0)) ; # R. J. Mathar, Mar 07 2016

RealDigits[Erf[Sqrt[2]], 10, 50][[1]] (* G. C. Greubel, Oct 19 2017 *)

Equals erf(sqrt(2)).

A249060 Column 1 of the triangular array at A249057.

Original entry on oeis.org

1, 4, 5, 24, 35, 192, 315, 1920, 3465, 23040, 45045, 322560, 675675, 5160960, 11486475, 92897280, 218243025, 1857945600, 4583103525, 40874803200, 105411381075, 980995276800, 2635284526875, 25505877196800, 71152682225625, 714164561510400, 2063427784543125

Offset: 0

Clark Kimberling, Oct 20 2014

			First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 4, a(2) = 5.

Clark Kimberling, Table of n, a(n) for n = 0..100

Cf. A178647, A249057, A309491.

z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t]; v1 = Flatten[CoefficientList[u, x]]; (* A249057 *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)

f(n) = if (n, x + (n + 3)/f(n-1), 1);
a(n) = polcoef(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

From Derek Orr, Oct 21 2014: (Start)
a(2*n) = (2*n+3)*(2*n+1)!!/3, for n > 0.
a(2*n+1) = (n+2)!*2^(n+1), for n > 0.
For n > 2, if n is even, a(n)/[(n+1)*(n-1)*(n-3)*...*7*5] = n + 3 and if n is odd, a(n)/[(n+1)*(n-1)*(n-3)*...*6*4] = n + 3. (End)
a(n) = gcd_2((n+3)!,(n+3)!!), where gcd_2(b,c) denotes the second-largest common divisor of non-coprime integers b and c, as defined in A309491. - Lechoslaw Ratajczak, Apr 15 2021
D-finite with recurrence: a(n) - (3+n)*a(n-2) = 0. - Georg Fischer, Nov 25 2022
Sum_{n>=0} 1/a(n) = 3*sqrt(e*Pi/2)*erf(1/sqrt(2)) + 2*sqrt(e) - 6, where erf is the error function. - Amiram Eldar, Dec 10 2022

A270712 Decimal expansion of the fraction of the normal distribution that falls within the 3 sigma error bars.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 22 2016

			0.997300203936739810946696370...

Wikipedia, 68-95-99.7 rule
Wikipedia, Normal distribution

Cf. A178647 (1sigma), A110894 (2sigma).

Maple
```
erf(3/sqrt(2)) ; evalf(%) ;
```

RealDigits[Erf[3/Sqrt[2]], 10, 120][[1]] (* Amiram Eldar, May 24 2023 *)

1 - erfc(3/sqrt(2)) \\ Michel Marcus, Mar 22 2016

A219337 Rounded frequency of population with score higher than mean +- n standard deviations.

Original entry on oeis.org

2, 6, 44, 741, 31574, 3488556, 1013594692, 781364430891, 1607468795310696, 8860626201053673385, 131236127104980388766388, 5233794723805693339116465076, 562910255724699183203714725974687, 163474435977817298005300626019111283694

Offset: 0

Joost de Winter, Nov 18 2012

			For example, 1 in 2 people have IQ greater than 100, about 1 in 6 people have IQ greater than 115, about 1 in 44 have IQ greater than 130, etc. (assuming normal IQ distribution with mean of 100 and standard deviation of 15)

Wikipedia, 68-95-99.7 rule

Cf. A178647.

MATLAB
```
round(2/(1-erf(n/sqrt(2))))
```

Table[Round[2/(1 - Erf[n/Sqrt[2]])], {n, 0, 15}] (* T. D. Noe, Dec 10 2012 *)

a(n)=round(2/erfc(n/sqrt(2))) \\ Charles R Greathouse IV, Dec 10 2012

Extended by T. D. Noe, Dec 10 2012

A379853 Decimal expansion of the fraction of a population falling beyond +- 1 standard deviation of the mean, assuming a normal distribution.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2025

			0.3173105078629141028295349087359241550441740665...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 40, table 40:7:1 at page 387.

Eric Weisstein's World of Mathematics, Erfc.

Cf. A178647, A239382.

RealDigits[Erfc[1/Sqrt[2]], 10, 100][[1]]

Equals 1 - A178647 = 2*A239382. - Hugo Pfoertner, Jan 04 2025

cp's OEIS Frontend

A051577 a(n) = (2*n + 3)!!/3 = A001147(n+2)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A110894 Decimal expansion of error function of square root of 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A249060 Column 1 of the triangular array at A249057.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A270712 Decimal expansion of the fraction of the normal distribution that falls within the 3 sigma error bars.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A219337 Rounded frequency of population with score higher than mean +- n standard deviations.