A092676 -id:A092676 - cp's OEIS frontend

A002067 a(n) = Sum_{k=0..n-1} binomial(2n,2k)a(k)a(n-k-1).

1, 1, 7, 127, 4369, 243649, 20036983, 2280356863, 343141433761, 65967241200001, 15773461423793767, 4591227123230945407, 1598351733247609852849, 655782249799531714375489, 313160404864973852338669783, 172201668512657346455126457343, 108026349476762041127839800617281

Offset: 0

N. J. A. Sloane

Number of increasing rooted triangular cacti of 2n+1 nodes. (In an increasing rooted graph, nodes are numbered and numbers increase as you move away from the root.)
a(n) is (2n)!/2^n times the n-th coefficient in the series for inverf(2x/sqrt(Pi)). - Paul Barry, Apr 12 2010
Number of ordered bilabeled increasing trees with 2n labels. - Markus Kuba, Nov 17 2014
Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 8/Pi. - Vaclav Kotesovec, Nov 19 2014
From David Callan, Jul 21 2017: (Start)
Conjectures:
a(n) is the Hafnian of the triangular array (u(i,j))_{1 <= i < j <= 2n} with u(i,j)=i. The Hafnian is the same as the Pfaffian except without the alternating signs just as the permanent of a matrix is the determinant without the signs.
a(n) is the total weight of Dyck n-paths with weight defined as follows. Given a Dyck path, for each upstep, record its position in the path and the height of its upper endpoint; then multiply together all of these positions and heights. For example, the Dyck 4-path P = UUDUUDDD has upsteps in positions 1,2,4,5 ending at heights 1,2,2,3 respectively, and hence weight(P) = 480. (In fact the positions determine the heights because, for the k-th upstep, position + height = 2k.) (End)

			E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 127*x^6/6! + 4369*x^8/8! + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, cf. Chapter 5.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470. [See Eq. 1.3 and Section 6.]
D. Dominici, Asymptotic analysis of the derivatives of the inverse error function, arXiv:math/0607230 [math.CA], 2006-2007.
A. J. E. M. Janssen, Analysis of a constrained initial value for an ODE arising in the study of a power-flow model, Eindhoven Univ. Tech. (Netherlands, 2023).
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
Wikipedia, Error Function
Index entries for sequences related to cacti

The sequence of fractions A092676/A132467 is closely related.

Periods: A122149, A122159.

a:=proc(n) option remember; if n <= 0 then RETURN(1); else RETURN( add( binomial(2*n,2*k)*a(k)*a(n-k-1), k=0..n-1 ) ); fi; end;

max = 16; se = Series[ InverseErf[ 2*x/Sqrt[Pi] ], {x, 0, 2*max+1} ]; a[n_] := (2n+1)!/2^n*Coefficient[ se, x, 2*n+1]; Table[ a[n], {n, 0, max} ] (* Jean-François Alcover, Mar 07 2012, after Paul Barry *)

/* E.g.f. A(x) = exp( Integral A(x) * Integral A(x) dx dx ): */
{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A * intformal( A + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", ")) \\ Paul D. Hanna, Jun 02 2015

/* By definition: */
{a(n) = if(n==0,1,sum(k=0,n-1, binomial(2*n,2*k)*a(k)*a(n-k-1)))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jun 02 2015

a(n) = b(2n+1), where e.g.f. of b satisfies B'(x)=exp(B(x)^2/2).
a(n) = (2n)! * A092676(n) / (2^n*A092677(n)). - Paul Barry, Apr 12 2010
a(n) = 1/2^n * A026944(n+1). Let D denote the operator g(x) -> (1/sqrt(2))*d/dx(exp(x^2)*g(x)). Then a(n) = D^(2*n)(1) evaluated at x = 0. - Peter Bala, Sep 08 2011
E.g.f. B(x)=Sum_{n>=1} a(n-1)*x^(2*n)/(2*n)! satisfies differential equation B''(x) - B(x)*B''(x) - 1 = 0, B'(0)=1/2. - Vladimir Kruchinin, Aug 12 2019
E.g.f. satisfies: A(x) = exp( Integral A(x)*B(x) dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and B(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!, and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015 [formula revised by Paul D. Hanna, Jul 06 2024 following a suggestion from Petros Hadjicostas]

Alternate description, formula and comment from Christian G. Bower

New definition and more terms from Vladeta Jovovic, Oct 22 2005

A092677 Denominators of coefficients in the series for inverf(2x/sqrt(Pi)).

Original entry on oeis.org

1, 3, 30, 630, 22680, 178200, 97297200, 10216206000, 198486288000, 237588086736000, 49893498214560000, 1803293578326240000, 222759794969712000000, 1329207696584271504000000

Offset: 1

Eric W. Weisstein, Mar 02 2004

nonn

Differs from A007019(n) at n = 6, 9, 12, ....

			inverf(2x/sqrt(Pi)) = x + x^3/3 + 7x^5/30 + 127x^7/630 + 4369x^9/22680 + 34807x^11/178200 + ...

G. C. Greubel, Table of n, a(n) for n = 1..235
G. Alkauskas, Algebraic and abelian solutions to the projective translation equation, arXiv preprint arXiv:1506.08028 [math.AG], 2015-2016; Aequationes Math. 90 (4) (2016), 727-763.
J. M. Blair, C. A. Edwards and J. H. Johnson, Rational Chebyshev approximations for the inverse of the error function, Math. Comp. 30 (1976), 827-830.
L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470.
Eric Weisstein, Mathematica program and first 50 terms of the series
Eric Weisstein's World of Mathematics, Inverse Erf

Cf. A007019, A092677.

Denominator[CoefficientList[Series[InverseErf[2*x/Sqrt[Pi]], {x, 0, 50}],
x]][[2 ;; ;; 2]] (* G. C. Greubel, Jan 09 2017 *)

A122551 Denominators of the coefficients of the series for InverseErf(x).

Original entry on oeis.org

2, 24, 960, 80640, 11612160, 2554675200, 797058662400, 334764638208000, 182111963185152000, 124564582818643968000, 104634249567660933120000, 105889860562472864317440000, 127067832674967437180928000000

Offset: 0

Marcus Blackburn (marcus.blackburn(AT)dial.pipex.com), Sep 20 2006

Note: the term in x^11 in the series expansion above has a common factor of 7 between the numerator and denominator and is usually written 34807/364953600. The common factor of 7 occurs at n=6, 9, 12, etc. The sequence of the coefficients can be generated by combining this series with A002067.

			InverseErf(x) = (1/2*sqrt(Pi))*x + (1/24*Pi^(3/2))*x^3 + (7/960*Pi^(5/2))*x^5 + (127/80640*Pi^(7/2))*x^7 + (4369/11612160*Pi^(9/2))*x^9 + (243649/2554675200*Pi^(11/2))*x^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..210

Cf. A002067, A092676.

denominators:=[seq((2*n+1)!*2^(n+1),n=0..14)]; a:=proc(n) if(n < 2) then RETURN(1) fi; sum('binomial(2*n,2*k)*a(k)*a(n-k-1)','k'=0..n-1); end; numerators:=[seq(a(n),n=0..14)];

Table[(2*n + 1)!*2^(n + 1), {n,0,25}] (* G. C. Greubel, Mar 19 2017 *)

for(n=0,25, print1((2*n+1)!*2^(n+1), ", ")) \\ G. C. Greubel, Mar 19 2017

a(n) = (2*n+1)!*2^(n+1).
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=0} 1/a(n) = sinh(1/sqrt(2))/sqrt(2).
Sum_{n>=0} (-1)^n/a(n) = sin(1/sqrt(2))/sqrt(2). (End)

cp's OEIS Frontend

A132467 Denominators associated with Taylor series expansion of inverse error function. See A092676 for numerators and further information.

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A002067 a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).

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A092677 Denominators of coefficients in the series for inverf(2x/sqrt(Pi)).

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Mathematica

A122551 Denominators of the coefficients of the series for InverseErf(x).

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Author

Keywords

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Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A002067 a(n) = Sum_{k=0..n-1} binomial(2n,2k)a(k)a(n-k-1).