A009564 -id:A009564 - cp's OEIS frontend

A103639 a(n) = Product_{i=1..2n} (2i+1).

1, 15, 945, 135135, 34459425, 13749310575, 7905853580625, 6190283353629375, 6332659870762850625, 8200794532637891559375, 13113070457687988603440625, 25373791335626257947657609375, 58435841445947272053455474390625, 157952079428395476360490147277859375

Offset: 0

Ralf Stephan, Feb 18 2005

			Sequence starts 1, 1*3*5, 1*3*5*7*9, 1*3*5*7*9*11*13, ...

G. C. Greubel, Table of n, a(n) for n = 0..175

Bisection of the double factorials A001147.
Cf. A102992.
Odd part of A024343 and A009564.

[(n+1)*Factorial(2*n+1)*Catalan(2*n+1)/4^n: n in [0..20]]; // G. C. Greubel, Jan 29 2022

A103639 := n -> pochhammer(1/2,2*n+1)*2^(2*n+1):
seq(A103639(n), n=0..11); # Peter Luschny, Dec 19 2012

Table[(4n+1)!!, {n, 0, 15}] (* Vladimir Reshetnikov, Nov 03 2015 *)

vector(20, n, n--; prod(i=1, 2*n, 2*i+1)) \\ Altug Alkan, Nov 04 2015

def A103639(n):
    return falling_factorial(4*n+2,2*n+1)*2^(-1-2*n)
print([A103639(n) for n in (0..11)])  # Peter Luschny, Dec 14 2012

a(n) = (4*n+2)! / (2 * 4^n * (2*n+1)! ).
E.g.f.: sinh(x^2/2) = x^2/2! + 15*x^6/6! + 945*x^10/10! +...
a(n+1) = (4*n-1)*(4*n+1)*a(n), a(0) = 1.
a(n) = (4*n+1)!!. - Vladimir Reshetnikov, Nov 03 2015
a(n) = denominator((-3/2 - 2*n)!/sqrt(Pi)). - Peter Luschny, Jun 21 2020
D-finite with recurrence a(n) -(4*n-1)*(4*n+1)*a(n-1) = 0. - R. J. Mathar, Jun 06 2025
Sum_{n>=0} 1/a(n) = (sqrt(Pi/2)/2) * (e * erf(1/sqrt(2)) + erfi(1/sqrt(2)) / e). - Amiram Eldar, Aug 31 2025

A024343 Expansion of e.g.f. sin(x^2) in powers of x^(4*n + 2).

Original entry on oeis.org

2, -120, 30240, -17297280, 17643225600, -28158588057600, 64764752532480000, -202843204931727360000, 830034394580628357120000, -4299578163927654889881600000, 27500101936481280675682713600000

Offset: 0

R. H. Hardin

Absolute values are coefficients of expansion of sinh(x^2).

G. C. Greubel, Table of n, a(n) for n = 0..175

Bisection of A001813.

Cf. A009564.

F:=Factorial;; [(-1)^n*F(4*n+2)/F(2*n+1) : n in [0..30]]; // G. C. Greubel, Jan 29 2022

Table[(-1)^n*(2*n+1)!*Binomial[4*n+2, 2*n+1], {n,0,30}] (* G. C. Greubel, Jan 29 2022 *)

a(n)=polcoeff(serlaplace(sin(x^2)),4*n+2)

PARI
```
a(n)=(-1)^n*(4*n+2)!/(2*n+1)!
```

f=factorial; [(-1)^n*f(4*n+2)/f(2*n+1) for n in (0..30)] # G. C. Greubel, Jan 29 2022

a(n) = (-1)^n * (4*n+2)! / (2*n+1)!.
E.g.f.: [x^(4*n+2)] sin(x^2)
a(n) = 2 * A009564(n). - Sean A. Irvine, Jul 01 2019
From Amiram Eldar, Sep 02 2025: (Start)
Sum_{n>=0} 1/a(n) = sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(Pi)/4) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)

Extended with signs by Olivier Gérard, Mar 15 1997
Edited by Ralf Stephan, Mar 25 2004
Name edited by Michel Marcus, Jul 01 2019

A220371 a(n) = Product_{i=1..2n} (4i+2)*A060818(n).

Original entry on oeis.org

1, 60, 30240, 17297280, 70572902400, 112634352230400, 518118020259840000, 1622745639453818880000, 53122201253160214855680000, 275173002491369912952422400000, 3520013047869603926487387340800000, 27244900990510734391012378017792000000

Offset: 0

Peter Luschny, Dec 13 2012

nonn

Cf. A009564, A060818, A103639, A193365.

a := proc(n): denom(binomial(1/2, iquo(n,2)))*product((4*i+2), i=1..2*n) end: seq(a(n), n=0..11); # Johannes W. Meijer, Dec 21 2012

a[n_] := 2^(2n)*Product[2i+1, {i, 1, 2n}]*GCD[n!, 2^n]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Dec 21 2012 *)

@CachedFunction
def A005187(n): return A005187(n//2) + n if n > 0 else 0
def A220371(n): return mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
[A220371(n) for n in range(12)]

a(n) = |A009564(n)|*A060818(n).

a(n) = 4*A193365(n)*a(n-1) with a(0) = 1. - Johannes W. Meijer, Dec 21 2012

cp's OEIS Frontend

A103639 a(n) = Product_{i=1..2n} (2i+1).

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A024343 Expansion of e.g.f. sin(x^2) in powers of x^(4*n + 2).

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PARI

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A220371 a(n) = Product_{i=1..2n} (4i+2)*A060818(n).

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cp's OEIS Frontend

A103639 a(n) = Product_{i=1..2*n} (2*i+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A024343 Expansion of e.g.f. sin(x^2) in powers of x^(4*n + 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

Extensions

A220371 a(n) = Product_{i=1..2*n} (4*i+2)*A060818(n).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A103639 a(n) = Product_{i=1..2n} (2i+1).

A220371 a(n) = Product_{i=1..2n} (4i+2)*A060818(n).