A047688 -id:A047688 - cp's OEIS frontend

A002017 Expansion of e.g.f. exp(sin(x)).

1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, 5973, 309376, 1237173, -2917888, -52635599, -163782656, 1126610929, 12716052480, 20058390573, -495644917760, -3920482183827, 4004259037184, 256734635981833, 1359174582304768

Offset: 0

N. J. A. Sloane

Number of set partitions of 1..n into odd parts with an even number of parts of size == 3 (mod 4), minus the number of such partitions with an odd number of parts of size == 3 (mod 4). - Franklin T. Adams-Watters, Apr 29 2010

			For n=6, there are 6 partitions with part sizes [5,1], 10 with sizes [3^2], 20 with sizes [3,1^3], and 1 with sizes [1^6]; 6 + 10 - 20 + 1 = -3. - _Franklin T. Adams-Watters_, Apr 29 2010

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
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..100
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|.
Cf. A003724. - Franklin T. Adams-Watters, Apr 29 2010
Cf. A047687, A047688 for numerators & denominators of the series of exp(sin(x)) at x = 0.

max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* Jean-François Alcover, Jun 26 2013 *)

a(n):=2*sum((sum((2*i-n+2*j)^n*binomial(n-2*j,i)*(-1)^(n-j-i),i,0,(n-2*j)/2))/(2^(n-2*j)*(n-2*j)!),j,0,(n-1)/2); /* Vladimir Kruchinin, Jun 10 2011 */

a(n):=if n=0 then 1 else (n-1)!*sum((-1)^(k)/(2*k)!*a(n-2*k-1)/(n-2*k-1)!,k,0,(n-1)/2); /* Vladimir Kruchinin, Feb 25 2015 */

my(x='x+O('x^33)); Vec(serlaplace(exp(sin(x)))) \\ Joerg Arndt, Apr 01 2017

a(n) = 2*Sum_{j=0..(n-1)/2} Sum_{i=0..(n-2*j)/2} (2*i-n+2*j)^n*C(n-2*j,i)*(-1)^(n-j-i)/(2^(n-2*j)*(n-2*j)!), n>0, a(0)=1. - Vladimir Kruchinin, Jun 10 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A003724. - Peter Bala, Dec 06 2011
E.g.f.: 1 + sin(x)/T(0), where T(k) = 4*k+1 - sin(x)/(2 + sin(x)/(4*k+3 - sin(x)/(2 + sin(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
E.g.f.: 2/Q(0), where Q(k) = 1 + 1/( 1 - sin(x)/( sin(x) - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
E.g.f.: E(0)-1, where E(k) = 2 + sin(x)/(2*k + 1 - sin(x)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = (n-1)!*Sum_{k=0..(n-1)/2} ((-1)^k/(2*k)!)*a(n-2*k-1)/(n-2*k-1)!, a(0)=1. - Vladimir Kruchinin, Feb 25 2015

Extended with signs by Christian G. Bower, Nov 15 1998

A047687 Numerators of coefficients in Taylor series for exp(sin(x)).

Original entry on oeis.org

1, 1, 1, 0, -1, -1, -1, 1, 31, 1, -2951, -1, 181, 2417, 58913, -5699, -52635599, -19993, 1126610929, 3631, 27069353, -6050353, -118802490419, 47438, 11162375477471, 41478716501, -8529964147714967, -3818348299, -2610006147952249, 36698180928319

Offset: 0

N. J. A. Sloane

			1 + 1*x + (1/2)*x^2 - (1/8)*x^4 - (1/15)*x^5 - (1/240)*x^6 + (1/90)*x^7 + (31/5760)*x^8 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

M. F. Hasler, Table of n, a(n) for n = 0..499

Cf. A047688 (denominators), A002017 (e.g.f.).

Numerator[CoefficientList[Series[Exp[Sin[x]],{x,0,30}],x]] (* Harvey P. Dale, Jan 22 2015 *)

A047687_upto(N=99)=apply(numerator, Vec(exp(sin(x+O(x^N))))) \\ M. F. Hasler, Jun 07 2025

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A002017 Expansion of e.g.f. exp(sin(x)).

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