A092820 -id:A092820 - cp's OEIS frontend

A009775 Exponential generating function is tanh(log(1+x)).

0, 1, -1, 0, 6, -30, 90, 0, -2520, 22680, -113400, 0, 7484400, -97297200, 681080400, 0, -81729648000, 1389404016000, -12504636144000, 0, 2375880867360000, -49893498214560000, 548828480360160000, 0, -151476660579404160000

Offset: 0

R. H. Hardin

Brandon Humpert and Jeremy L. Martin, The Incidence Hopf Algebra of Graphs, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 517-526. [See Example 3.4]

Cf. A052277, A007019, A046979, A007415, A007452, A092820, A217260.

A009775(n)=polcoeff(tanh(log(1+x+O(x^n)*x)),n)*n!  \\ M. F. Hasler, Oct 10 2012

a(0) = 0, a(4n+3) = 0, a(n) = (-1)^[n == 2, 5, 8 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004
From Peter Bala, Nov 25 2011: (Start)
(1): a(n) = i*n!/2^(n+1)*{(i-1)^(n+1)-(-1-i)^(n+1)} for n>=1.
The function tanh(log(1+x)) is a disguised form of the rational function (x^2+2*x)/(x^2+2*x+2). Observe that
(2): (x^2+2*x)/(x^2+2*x+2) = d/dx[x - atan((x^2+2*x)/(2*x+2))].
Hence, with an offset of 1, the egf for this sequence is
(3): x - atan((x^2+2*x)/(2*x+2)) = x^2/2! - x^3/3! + 6*x^5/5!- 30*x^6/6! + 90*x^7/7! - ....
This sequence is closely related to the series reversion of the function E(x)-1, where E(x) = sec(x)+tan(x) is the egf for the sequence of zigzag numbers A000111. Under the change of variable x -> sec(x)+tan(x)-1 the rational function (x^2+2*x)/(2*x+2) transforms to tan(x). Hence atan((x^2+2*x)/(2*x+2)) is the inverse function of sec(x)+tan(x)-1.
Recurrence relation:
(4): 2*a(n)+2*n*a(n-1)+n*(n-1)*a(n-2) = 0 with a(1) = 1, a(2) = -1.
(End)

Extended with signs by Olivier Gérard, Mar 15 1997

A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

Original entry on oeis.org

0, 1, -1, 3, 0, -30, 90, -630, 0, 22680, -113400, 1247400, 0, -97297200, 681080400, -10216206000, 0, 1389404016000, -12504636144000, 237588086736000, 0, -49893498214560000, 548828480360160000, -12623055048283680000, 0, 3786916514485104000000

Offset: 0

N. J. A. Sloane

sign

J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.

Absolute values are essentially the same as A046979, where zeros are replaced by ones.
a(4n+2) = -(-1)^n*A052277(n), a(2n+1) = (-1)^[n/2]*A007019(n).
Cf. A046981, A007452, A009775, A092820.

a:= n-> (p-> `if`(p=0,0,1/p))(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

1/CoefficientList[Sin[x]/Exp[x] + O[x]^26, x] /. ComplexInfinity -> 0 // Quiet (* Jean-François Alcover, Feb 26 2019 *)

a(n) = [n mod 4 > 0] * (-1)^(n+1+[n/4]) * n!/2^[n/2]. - Ralf Stephan, Mar 06 2004

E.g.f.: sin(x)/exp(x) = x-x^2/(G(0)+x); G(k)=2k+1-x+x*(2k+1)/(4k+3-x+x^2*(4k+3)/( (2k+2)*(4k+5)-x^2+x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011

A046979 Denominators of Taylor series for exp(x)*sin(x).

Original entry on oeis.org

1, 1, 1, 3, 1, 30, 90, 630, 1, 22680, 113400, 1247400, 1, 97297200, 681080400, 10216206000, 1, 1389404016000, 12504636144000, 237588086736000, 1, 49893498214560000, 548828480360160000, 12623055048283680000, 1, 3786916514485104000000, 49229914688306352000000

Offset: 0

N. J. A. Sloane

			1*x + 1*x^2 + 1/3*x^3 - 1/30*x^5 - 1/90*x^6 - 1/630*x^7 + 1/22680*x^9 + 1/113400*x^10 + ...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Cf. A046978, A046981, A052277, A007019, A007452, A009775, A092820.

Essentially the same as absolute values of A007415.

a:= n-> denom(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

Denominator[CoefficientList[Series[Exp[x]Sin[x],{x,0,30}],x] ] (* Harvey P. Dale, Feb 14 2015 *)

a(n) = if (n % 4, n!/2^floor(n/2), 1); \\ Michel Marcus, Oct 12 2015

a(4n) = 1, a(n) = n!/2^floor(n/2).

A046981 Denominators of Taylor series for exp(x)*cos(x).

Original entry on oeis.org

1, 1, 1, 3, 6, 30, 1, 630, 2520, 22680, 1, 1247400, 7484400, 97297200, 1, 10216206000, 81729648000, 1389404016000, 1, 237588086736000, 2375880867360000, 49893498214560000, 1, 12623055048283680000, 151476660579404160000, 3786916514485104000000

Offset: 0

N. J. A. Sloane

			1+1*x-1/3*x^3-1/6*x^4-1/30*x^5+1/630*x^7+1/2520*x^8+1/22680*x^9-...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A046980, A046979, A052277, A007019, A007415, A009775, A092820.

Absolute values are essentially the same as A007452.

Denominator/@CoefficientList[Series[Exp[x]Cos[x],{x,0,30}],x] (* Harvey P. Dale, Jun 28 2011 *)

a(4n+2) = 1, a(n) = n!/2^[n/2].

A007452 Expand cos x / exp x and invert nonzero coefficients.

Original entry on oeis.org

1, -1, 0, 3, -6, 30, 0, -630, 2520, -22680, 0, 1247400, -7484400, 97297200, 0, -10216206000, 81729648000, -1389404016000, 0, 237588086736000, -2375880867360000, 49893498214560000, 0, -12623055048283680000

Offset: 0

N. J. A. Sloane

sign

Absolute values are essentially the same as A046981, where zeros are replaced by ones.

Cf. A052277, A007019, A046979, A007415, A009775, A092820.

a(4n+2) = 0, a(n) = (-1)^[n == 1, 4, 7 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004

A092830 Central Delannoy numbers that are primes.

Original entry on oeis.org

3, 13, 265729

Offset: 1

Eric W. Weisstein, Mar 06 2004

Apart from the first term this is a subsequence of A101167. - Reinhard Zumkeller, Dec 03 2004

Next term, if it exists, is > 10^(300). - Emeric Deutsch, Mar 01 2005

Eric Weisstein's World of Mathematics, Delannoy Number

Cf. A001850, A092820, A008288.

cp's OEIS Frontend

A009775 Exponential generating function is tanh(log(1+x)).

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A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

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A046979 Denominators of Taylor series for exp(x)*sin(x).

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Maple

Mathematica

PARI

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A046981 Denominators of Taylor series for exp(x)*cos(x).

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Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

A007452 Expand cos x / exp x and invert nonzero coefficients.

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Formula

A092830 Central Delannoy numbers that are primes.

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