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A002017 E.g.f. exp(sin(x)).
(Formerly M2709 N1086)
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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.

Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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 */

CROSSREFS

a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|.

Cf. A003724. - Franklin T. Adams-Watters, Apr 29 2010

Sequence in context: A146975 A046970 A058936 * A278292 A278957 A118582

Adjacent sequences:  A002014 A002015 A002016 * A002018 A002019 A002020

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified December 7 11:42 EST 2016. Contains 278874 sequences.