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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Comments

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

Examples

			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
		

References

• 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).

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

Maxima

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 */

Maxima

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 */

PARI

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

Formula

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

Extensions

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