A101485 -id:A101485 - cp's OEIS frontend

A009120 a(n) = (4n)!/(2n)!.

1, 12, 1680, 665280, 518918400, 670442572800, 1295295050649600, 3497296636753920000, 12576278705767096320000, 58102407620643984998400000, 335367096786357081410764800000, 2365008766537390138108713369600000, 20007974164906320568399715106816000000

Offset: 0

R. H. Hardin

Absolute value of the coefficients in the expansion of cos(x^2). - clarified by Muniru A Asiru, Jul 26 2018
Bisection of sequence A001813. - Gary W. Adamson, Jul 19 2011
Expansion of cosh(x^2) in powers of x^4. - G. C. Greubel, Jul 26 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A001813, A101485.

List([0..25],n->Factorial(4*n)/Factorial(2*n)); # Muniru A Asiru, Jul 26 2018

[Factorial(4*n)/Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Jul 20 2011

seq(coeff(series(factorial(n)*cosh(x^2), x,n+1),x,n),n=0..50,4); # Muniru A Asiru, Jul 27 2018

Table[(4n)!/(2n)!,{n,0,10}] (* or *) With[{nn=60},Abs[Take[ CoefficientList[ Series[ Cos[x^2],{x,0,nn}],x] Range[0,nn]!,{1,-1,4}]]] (* Harvey P. Dale, Mar 27 2012 *)

for(n=0, 20, print1((4*n)!/(2*n)!, ", ")) \\ G. C. Greubel, Jul 26 2018

my(x='x+O('x^120)); v=Vec(serlaplace(cosh(x^2))); vector(#v\4, n, v[4*n-3]) \\ G. C. Greubel, Jul 26 2018

a(n) = 4^n * A101485(n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(1/4)*exp(-1/4*sqrt(x))/(sqrt(Pi)*x^(3/4)) dx, n >= 0. - Karol A. Penson, Sep 19 2001
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^(2n), where M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ... (End)
Sum_{n>=0} 1/a(n) = 1 + (1/4) * exp(1/4) * sqrt(Pi) * erf(1/2) - (1/4) * exp(-1/4) * sqrt(Pi) * erfi(1/2), where erf is the error function and erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023

Extended by Olivier Gérard, Mar 01 1997

A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025, -34459425, 654729075, -13749310575, 316234143225, -7905853580625, 213458046676875, -6190283353629375, 191898783962510625, -6332659870762850625, 221643095476699771875, -8200794532637891559375, 319830986772877770815625

Offset: 0

Peter Luschny, Jan 06 2020

sign

G. C. Greubel, Table of n, a(n) for n = 0..400
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 25.

The equivalent for Stirling2 is A009235.
Cf. A001147, A101485, A103639, A130534.
Cf. A330799, A330800, A330802, A330803.

m:=30;
R:=PowerSeriesRing(Rationals(), m+2);
A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;
[A330797(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023

a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);
# Alternative:
arec := proc(n) option remember: if n = 0 then 1 else
(3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);
# Or:
gf := (1+2*x)^(1/2); ser := series(gf, x, 24);
seq(n!*coeff(ser, x, n), n=0..20);

a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 19 2021 *)
Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n,0,30}] (* G. C. Greubel, Sep 14 2023 *)

def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
[A330797(n) for n in (0..20)]

a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.
a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).
a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).
a(n) = n!*[x^n] (1+2*x)^(1/2).
D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).
a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).
a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).
a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).
a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).
Sum_{n>=0} 1/a(n) = 2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023
O.g.f.: 1+x*2F0(1/2,1;;-2*x). - R. J. Mathar, Aug 10 2025

A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

Original entry on oeis.org

1, 1, 3, 1, 15, 105, 1, 35, 945, 10395, 1, 63, 3465, 135135, 2027025, 1, 99, 9009, 675675, 34459425, 654729075, 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225, 1, 195, 36465, 6235515, 916620705, 105411381075, 7905853580625, 213458046676875

Offset: 0

Peter Luschny, May 05 2023

			[0] 1;
[1] 1,   3;
[2] 1,  15,   105;
[3] 1,  35,   945,   10395;
[4] 1,  63,  3465,  135135,   2027025;
[5] 1,  99,  9009,  675675,  34459425,   654729075;
[6] 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225;

Cf. A362848 (row sums), A000466 (column 1), A101485 (main diagonal).

T := (n, k) -> 4^k * GAMMA(n + k + 1/2) / GAMMA(n - k + 1/2):
seq(seq(T(n, k), k = 0..n), n = 0..7);

T[n_,k_]:=(2*(n+k)-1)!!/(2*(n-k)-1)!!;Flatten[Table[T[n,k],{n,0,7},{k,0,n}]] (* Detlef Meya, Oct 09 2023 *)

T(n ,k ) = (2*(n + k) - 1)!!/(2*(n - k) - 1)!!; 0 <= n <= k. - Detlef Meya, Oct 09 2023

cp's OEIS Frontend

A009120 a(n) = (4n)!/(2n)!.

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A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

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Magma

Maple

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A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

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Maple

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cp's OEIS Frontend

A009120 a(n) = (4*n)!/(2*n)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A009120 a(n) = (4n)!/(2n)!.