A103639 -id:A103639 - cp's OEIS frontend

A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

1, 5, 21, 715, -162877, 19840275, -7176079695, 1829885835675, -5009184735027165, 2216222559226679575, -2463196751104762933637, 1679951011110471133453965, -5519118103058048675551057049, 5373485053345792589762994345215, -12239617587594386225052760043303511

Offset: 0

Peter Luschny, Dec 27 2012

sign

Let N = 4*n+3 and A = sum_{k>=0} a(k)/(A123854(k)*N^(2*k)) then
C(n) ~ 8*4^n*A/(N*sqrt(N*Pi)), C(n) = (4^n/sqrt(Pi))*(Gamma(n+1/2)/ Gamma(n+2)) the Catalan numbers A000108.
The asymptotic expansion of the Catalan numbers considered here is based on the Taylor expansion of square root of the sine cardinal. This asymptotic series involves only even powers of N, making it more efficient than the asymptotic series based on Stirling's approximation to the central binomial which involves all powers (see for example: D. E. Knuth, 7.2.1.6 formula (16)). The series is discussed by Kessler and Schiff but is included as a special case in the asymptotic expansion given by J. L. Fields for quotients Gamma(x+a)/Gamma(x+b) and discussed by Y. L. Luke (p. 34-35), apparently overlooked by Kessler and Schiff.

			With N = 4*n+3 the first few terms of A are A = 1 + 5/(4*N^2) + 21/(32*N^4) + 715/(128*N^6) - 162877/(2048*N^8) + 19840275/(8192*N^10). With this A C(n) = round(8*4^n*A/(N*sqrt(N*Pi))) for n = 0..39 (if computed with sufficient numerical precision).

Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees—History of Combinatorial Generation, 2006.
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

Peter Luschny, Table of n, a(n) for n = 0..99
J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.

The logarithmic version is A220422. Appears in A193365 and A220466.

Cf. A220412.

A220002 := proc(n) local s; s := n -> `if`(n > 0, s(iquo(n,2))+n, 0);
(-1)^n*mul(4*i+2, i = 1..2*n)*2^s(iquo(n,2))*coeff(taylor(sqrt(sin(x)/x), x,2*n+2), x, 2*n) end: seq(A220002(n), n = 0..14);
# Second program illustrating J. L. Fields expansion of gamma quotients.
A220002 := proc(n) local recF, binSum, swing;
binSum := n -> add(i,i=convert(n,base,2));
swing := n -> n!/iquo(n, 2)!^2;
recF := proc(n, x) option remember; `if`(n=0, 1, -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*recF(n-2*k-2,x),k=0..n/2-1)) end: recF(2*n,-1/4)*2^(3*n-binSum(n))*swing(4*n+1) end:

max = 14; CoefficientList[ Series[ Sqrt[ Sinc[x]], {x, 0, 2*max+1}], x^2][[1 ;; max+1]]*Table[ (-1)^n*Product[ (2*k+1), {k, 1, 2*n}], {n, 0, max}] // Numerator (* Jean-François Alcover, Jun 26 2013 *)

length = 15; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
def A005187(n): return A005187(n//2) + n if n > 0 else 0
def A220002(n):
    P = mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
    return (-1)^n*P*T.coefficient(x, 2*n)
[A220002(n) for n in range(length)]

# Second program illustrating the connection with the Euler numbers.
def A220002_list(n):
    S = lambda n: sum((4-euler_number(2*k))/(4*k*x^(2*k)) for k in (1..n))
    T = taylor(exp(S(2*n+1)),x,infinity,2*n-1).coefficients()
    return [t[0].numerator() for t in T][::-1]
A220002_list(15)

Let [x^n]T(f(x)) denote the coefficient of x^n in the Taylor expansion of f(x) then r(n) = (-1)^n*prod_{i=1..2n}(2i+1)*[x^(2*n)]T(sqrt(sin(x)/x)) is the rational coefficient of the asymptotic expansion (in N=4*n+3) and a(n) = numerator(r(n)) = r(n)*2^(3*n-bs(n)), where bs(n) is the binary sum of n (A000120).
Also a(n) = numerator([x^(2*n)]T(exp(S))) where S = sum_{k>=1}((4-E(2*k))/ (4*k)*x^(2*k)) and E(n) the Euler numbers A122045.
Also a(n) = sf(4*n+1)*2^(3*n-bs(n))*F_{2*n}(-1/4) where sf(n) is the swinging factorial A056040, bs(n) the binary sum of n and F_{n}(x) J. L. Fields' generalized Bernoulli polynomials A220412.
In terms of sequences this means
r(n) = (-1)^n*A103639(n)*A008991(n)/A008992(n),
a(n) = (-1)^n*A220371(n)*A008991(n)/A008992(n).
Note that a(n) = r(n)*A123854(n) and A123854(n) = 2^A004134(n) = 8^n/2^A000120(n).
Formula from Johannes W. Meijer:
a(n) = d(n+1)*A098597(2*n+1)*(A008991(n)/A008992(n)) with d(1) = 1 and
d(n+1) = -4*(2*n+1)*A161151(n)*d(n),
d(n+1) = (-1)^n*2^(-1)*(2*(n+1))!*A060818(n)*A048896(n).

A273889 a(n) = ((4n-3)!! + (4n-2)!!) / (4n-1).

Original entry on oeis.org

1, 9, 435, 52017, 11592315, 4152126825, 2182133628675, 1581940549814625, 1512952069890336075, 1845586177840605209625, 2796710279417971723681875, 5153962250373844341910100625, 11351091844757135191108560046875, 29444207228221006416048397134215625, 88848552445321896564985597922269171875

Offset: 1

Jiangshan Sun, Brian Cheung, Jason Y.S. Chiu, Hong-Chang Wang, Jun 02 2016

nonn

			a(1) = (1 + 2)/3 = 1;
a(2) = (1*3*5 + 2*4*6)/7 = 9;
a(3) = (1*3*5*7*9 + 2*4*6*8*10)/11 = 435.

Hong-Chang Wang, Table of n, a(n) for n = 1..100
Brian Cheung, Proof that (4n-1) divides (4n-3)!! + (4n-2)!!
Hong-Chang Wang, Bridan's blog (in Chinese)

Cf. A000165, A001147, A006882, A004767, A103639.

B[n_, k_] := (Product[k (i - 1) + 1, {i, 2 n - 1}] + Product[k (i - 1) + 2, {i, 2 n - 1}])/(2 k (n - 1) + 3); Table[B[n, 2], {n, 15}] (* Michael De Vlieger, Jun 10 2016 *)
Table[((4n-3)!!+(4n-2)!!)/(4n-1),{n,20}] (* Harvey P. Dale, Mar 08 2018 *)

for n in range(1,101):
    if n == 1:
        a = 1
        b = 2
    else:
        a = a*(4*n-5)*(4*n-3)
        b = b*(4*n-4)*(4*n-2)
    c = (a+b)/(4*n-1)
    print(str(n)+" "+str(c))

A009564 E.g.f. sin(x^2)/2, coefficients of x^(4*n + 2).

Original entry on oeis.org

1, -60, 15120, -8648640, 8821612800, -14079294028800, 32382376266240000, -101421602465863680000, 415017197290314178560000, -2149789081963827444940800000, 13750050968240640337841356800000, -106425394494182556214892101632000000, 980390734080409707851586040233984000000

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..182

Cf. A001813, A103639, A024343, A075069.

[(-1)^n*Factorial(2+4*n)/(2*Factorial(1+2*n)): n in [0..20]]; // Vincenzo Librandi, Dec 22 2015

seq(i!*coeff(series(sin(x^2)/2,x,4*i+4),x,i),i=2..54,4); # Peter Luschny, Dec 14 2012

nmax = 12; coes = CoefficientList[ Series[ Sin[x^2]/2, {x, 0, 4*nmax + 2}], x]; a[n_] := coes[[4*n + 3]]*(4*n + 2)!; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 14 2012 *)
Table[(-1)^n (2 + 4 n)!/(2 (1 + 2 n)!), {n, 0, 25}] (* Vincenzo Librandi, Dec 22 2015 *)

a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!); \\ Altug Alkan, Dec 22 2015

def A009564(n):
    return falling_factorial(4*n+2,2*n+1)/(2*(-1)^n)
[A009564(n) for n in (0..12)]  # Peter Luschny, Dec 14 2012

a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!) = (-1)^n*A001813(2*n+1)/2. - Robert Israel, Dec 21 2015
From Amiram Eldar, Sep 02 2025: (Start)
a(n) = A024343(n)/2.
Sum_{n>=0} 1/a(n) = sqrt(2*Pi) * (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)/2) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)

Extended with signs Mar 1997

Definition corrected and terms a(10)-a(12) from Peter Luschny, Dec 14 2012

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

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

A376922 Variance of n-th power of a standard normal random variable.

Original entry on oeis.org

1, 2, 15, 96, 945, 10170, 135135, 2016000, 34459425, 653836050, 13749310575, 316126087200, 7905853580625, 213439785208650, 6190283353629375, 191894675132160000, 6332659870762850625, 221641908024728441250, 8200794532637891559375, 319830558102716120460000

Offset: 1

Adam M. Scherlis, Oct 10 2024

nonn

The variance of the n-th sample moment is exactly a(n) / k for sample size k.

For a non-standard normal r.v. X ~ N(0, sigma^2), Var(X^n) = a(n) sigma^(2n).

			If Z ~ N(0, 1), then Var(Z) = 1, Var(Z^2) = 2, Var(Z^3) = 15, etc.

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics Volume 1, Charles Griffin & Company, 1963, page 229.

Cf. A001147, A103639, A123023.

a(n) = M(2n) - M(n)^2, where M(n) = A123023(n) are the moments of N(0, 1).

cp's OEIS Frontend

A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

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A273889 a(n) = ((4n-3)!! + (4n-2)!!) / (4n-1).

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A009564 E.g.f. sin(x^2)/2, coefficients of x^(4*n + 2).

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Magma

Maple

Mathematica

PARI

Sage

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Extensions

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

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Magma

Maple

Mathematica

SageMath

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

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Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A376922 Variance of n-th power of a standard normal random variable.

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Author

Keywords

Comments

Examples

References

Crossrefs

Formula

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