A002067 -id:A002067 - cp's OEIS frontend

A122149 Period of A002067 mod n.

1, 1, 1, 4, 8, 1, 3, 4, 9, 8, 10, 4, 24, 3, 8, 4, 32, 9, 18, 8, 3, 10, 22, 4

Offset: 1

N. J. A. Sloane, Aug 06 2008

			A002067 mod 5 is 1, 1, 2, 2, 4, 4, 3, 3, 1, 1, 2, 2, 4, 4, 3, 3, 1, 1, ... with period 8.

Cf. A002067, A122159.

max = 40; se = Series[InverseErf[2*x/Sqrt[Pi]], {x, 0, 2*max + 1}]; a[n_] := (2*n + 1)!/2^n*Coefficient[se, x, 2*n + 1]; A002067 = Table[a[n], {n, 0, max}]; period[lst_List] := Catch[lg = If[Length[lst] <= 5, 2, 5]; lst1 = lst[[1 ;; lg]]; km = Length[lst] - lg; Do[ If[lst1 == lst[[k ;; k+lg-1]], Throw[k-1]]; If[k == km, Throw[0]], {k, 2, km}]]; Table[period[Mod[A002067, n] // Reverse], {n, 1, 24}] (* Jean-François Alcover, Dec 17 2012 *)

Extended to 24 terms by Jean-François Alcover, Dec 17 2012

A122159 Period of A002067 modulo prime(n).

Original entry on oeis.org

1, 1, 8, 3, 10, 24, 32, 18, 22, 56, 30, 72, 80, 42, 23, 104, 29, 120, 66, 70, 144, 39, 41, 176, 192, 200, 51, 53, 216, 224, 63, 130, 272, 69, 296, 150, 312, 162, 166, 344, 178, 360, 95, 384, 392, 99, 105, 222, 226, 456, 464, 238, 480, 125, 512, 131, 536, 270, 552

Offset: 1

N. J. A. Sloane, Aug 06 2008

nonn

			A002067 modulo 5 is 1, 1, 2, 2, 4, 4, 3, 3, 1, 1, 2, 2, 4, 4, 3, 3, 1, 1, ... with period 8.

Cf. A002067, A122149.

max = 100; se = Series[InverseErf[2*x/Sqrt[Pi]], {x, 0, 2*max + 1}]; a[n_] := (2 n + 1)!/2^n*Coefficient[se, x, 2*n + 1]; A002067 = Table[a[n], {n, 0, max}]; period[lst_List] := Catch[lg = If[Length[lst] <= 5, 2, 5]; lst1 = lst[[1 ;; lg]]; km = Length[lst] - lg; Do[If[lst1 == lst[[k ;; k + lg - 1]], Throw[k - 1]]; If[k == km, Throw[0]], {k, 2, km}]]; Table[ period[Mod[A002067, Prime[n]] // Reverse] , {n, 1, 15}] (* Jean-François Alcover, Dec 17 2012 *)

a(n) = A122149(A000040(n)).

a(9)-a(15) from Jean-François Alcover, Dec 17 2012

More terms from Jinyuan Wang, Jul 30 2022

A007019 a(n) = (2n+1)! / 2^n.

Original entry on oeis.org

1, 3, 30, 630, 22680, 1247400, 97297200, 10216206000, 1389404016000, 237588086736000, 49893498214560000, 12623055048283680000, 3786916514485104000000, 1329207696584271504000000, 539658324813214230624000000, 250941121038144617240160000000, 132496911908140357902804480000000

Offset: 0

N. J. A. Sloane

Denominators of coefficients of the Taylor series of sinh(sqrt(2*x))/(sqrt(2*x)). - J. Zurita (jrzurita(AT)inaoep.mx), Dec 01 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..90
L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470.
R. Flórez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12 (2012), no. 1, 83-96.
Z. Usiskin, Letter to N. J. A. Sloane, Oct. 1991
Eric Weisstein's World of Mathematics, Inverse Erf

Numerators: A002067, erf(x): A007680.

[Factorial(2*n+1)/2^n: n in [0..25]]; // Vincenzo Librandi, May 14 2011

a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*(2*n-1)*(n-1) od: seq(a[n], n=1..14); # Zerinvary Lajos, Mar 08 2008

Table[(2n+1)!/2^n,{n,0,20}] (* Harvey P. Dale, May 13 2011 *)

a(n) = (2*n+1)!/2^n; \\ Altug Alkan, Aug 27 2018

sin(x)*cosh(x) = Sum_{n>=0} (-1)^floor(n/2)*x^(2n+1)/a(n). - Benoit Cloitre, Feb 02 2002
a(n) = Product_{k=0..n-1} (A000217(n+1) - A000217(k)). - Anton Zakharov, Sep 14 2016
a(n) ~ sqrt(Pi)*2^(n+2)*n^(2*n+3/2)/exp(2*n). - Ilya Gutkovskiy, Sep 14 2016
a(n) = Product_{j=1..n} T(2j) (where T(k) is the k-th triangular number). For example: a(3) = T(2)*T(4)*T(6) (that is, 630 = 3*10*21). - Rigoberto Florez, Aug 26 2018
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = sinh(sqrt(2))/sqrt(2).
Sum_{n>=0} (-1)^n/a(n) = sin(sqrt(2))/sqrt(2). (End)

A092676 Numerators of coefficients in the series for inverf(2x/sqrt(Pi)).

Original entry on oeis.org

1, 1, 7, 127, 4369, 34807, 20036983, 2280356863, 49020204823, 65967241200001, 15773461423793767, 655889589032992201, 94020690191035873697, 655782249799531714375489, 44737200694996264619809969

Offset: 1

Eric W. Weisstein, Mar 02 2004

Differs from A002067(n) at n = 6, 9, 12, ....

Following Blair et al., we use the notation inverf() for the inverse of the error function.

			Inverf(2x/sqrt(Pi)) = x + x^3/3 + 7x^5/30 + 127x^7/630 + 4369x^9/22680 + 34807x^11/178200 + ...
The first few coefficients are 1, 1, 7/6, 127/90, 4369/2520, 34807/16200, 20036983/7484400, 2280356863/681080400, ...

G. C. Greubel, Table of n, a(n) for n = 1..230
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
Wikipedia, Error Function

Cf. A002067, A092677, A052712. For denominators see A132467.

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

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

Edited by N. J. A. Sloane, Nov 15 2007

A026944 E.g.f. is inverse function to y -> integral from 0 to y of exp(-s^2).

Original entry on oeis.org

1, 2, 28, 1016, 69904, 7796768, 1282366912, 291885678464, 87844207042816, 33775227494400512, 16152024497964817408, 9402833148376976193536, 6546848699382209957269504, 5372168190357763804164005888, 5130820073307731596716765724672, 5642704273822755928641583754215424

Offset: 1

F. Chapoton, Mar 22 2000

nonn

The generating function is odd, so this list contains only the nonzero coefficients in the Taylor expansion.

Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 16/Pi. - Vaclav Kotesovec, Nov 19 2014

Vaclav Kotesovec, Table of n, a(n) for n = 1..220
D. Dominici, Asymptotic analysis of the derivatives of the inverse error function, arXiv:math/0607230 [math.CA], 2006-2007.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

Cf. A002067.

MakeTable[n_] := Select[CoefficientList[Series[InverseErf[2x/Sqrt[Pi]],{x,0,2n+1}],x] Table[k!, {k,0,2n+1}], # != 0 &]; MakeTable[15] (* Emanuele Munarini, Dec 17 2012 *)
nmax=20; c = ConstantArray[0,nmax]; c[[1]]=1; Do[c[[k+1]] = Sum[c[[m+1]]*c[[k-m]]/(m+1)/(2*m+1),{m,0,k-1}],{k,1,nmax-1}]; A026944=c*(2*Range[0,nmax-1])! (* Vaclav Kotesovec, Feb 25 2014 *)

f(n):=n!/2*coeff(taylor(2*inverse_erf(2*x/sqrt(%pi)),x,0,n),x,n); makelist(f(2*n+1),n,0,12); /* Emanuele Munarini, Dec 17 2012 */

v=Vec(serlaplace(serreverse(intformal(exp(-x^2)))));
vector(#v\2,n,v[2*n-1])  /* show terms */
/* Demonstration of Kruchinin's differential equation: */
default(seriesprecision,55); /* that many terms */
A=serreverse(intformal(exp(-x^2))); /* e.g.f. */
deriv(A)-exp(A^2)  /* gives O(x^57), i.e., zero up to order */

Nonzero constant terms of the polynomials P_{2n-1} in t defined by P_1=1, P_{n+1} = P'n+2*n*t*P_n.
E.g.f.: (1/2*sqrt(Pi)*erf)^{-1}(x).
a(n) = A002067(n-1) * 2^(n-1).
E.g.f. A(x) satisfies the differential equation A'(x) = exp(A(x)^2). - Vladimir Kruchinin, Jan 22 2011
Let D denote the operator g(x) -> d/dx(exp(x^2)*g(x)). Then a(n) = D^(2*n-2)(1) evaluated at x = 0. See [Dominici, Example 11]. - Peter Bala, Sep 08 2011

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)

A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

Original entry on oeis.org

1, 7, 3, 127, 123, 30, 4369, 6822, 3579, 630, 243649, 532542, 439899, 162630, 22680, 20036983, 56717781, 64697499, 37155267, 10735470, 1247400, 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200, 343141433761, 1427877062076, 2563294235106, 2572662311496, 1558544277681, 569674791180, 116270210700, 10216206000

Offset: 1

Matthew Miller, Dec 11 2018

It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.
It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.
It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.
It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.
This triangular array arises as the coefficients from terms in the inverse expansion of the function f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t evaluated at t = 0 for even values of the operation, using a method of Dominici's (nested derivatives, referenced below).
Without proof, appears to be related to computing the 'critical t-value' of Student's t-distribution. (conj.) Critical t-value t_(v, beta) is equal to: sqrt((v/(1-S^2)) - v) where S = (1/2)*Sum_{k>=1} (D^(2*k-2)[f](0)*(1/(2*k-1)!)*(B(1/2, v/2)*(1-2*beta))^(2*k-1)); where (1 - beta) is the confidence interval 'atta' (for a one-tailed distribution such that 'cumulative probability' = t_atta, where beta = 1-atta), x = 1 - (v/2), v: degrees of freedom, B(1/2, v/2) = gamma(1/2)*gamma(v/2)/gamma(1/2 + v/2), D^(2*k - 2)[f](0) is a polynomial function of 'x' whose coefficients are the terms of this sequence as computed using a method of Dominici's on f(t,x) with respect to t (referenced below).

			Given D^k[f]_(b) = (d/dt [f(t)*D^(k-1)[f](t)])_t = b where D^0[f](b) = 1, then for f(t,x) = (1/2)*(1 - t^2)^(-x) where f(0) = 1/2 one obtains: D^2[f]_(0) = -x/2, D^4[f]_(0) = (x/4)*(7*x - 3), D^6[f]_(0) = -(x/8)*(127*x^2 - 123*x + 30), etc., where b is an arbitrary constant.
Triangle begins:
           1;
           7,          3;
         127,        123,          30;
        4369,       6822,        3579,        630;
      243649,     532542,      439899,     162630,      22680;
    20036983,   56717781,    64697499,   37155267,   10735470,   1247400;
  2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
         ...

Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
Wikipedia, Student's t-distribution

Cf. A002067, A007019, A025035, A002105.

cp's OEIS Frontend

A122149 Period of A002067 mod n.

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A122159 Period of A002067 modulo prime(n).

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A007019 a(n) = (2n+1)! / 2^n.

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A092676 Numerators of coefficients in the series for inverf(2x/sqrt(Pi)).

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A026944 E.g.f. is inverse function to y -> integral from 0 to y of exp(-s^2).

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

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A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

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