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 A002017 E.g.f. exp(sin(x)). (Formerly M2709 N1086) 12
 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 MAPLE a:=series(exp(sin(x)), x=0, 26): seq(n!*coeff(a, x, n), n=0..25); # Paolo P. Lava, Mar 26 2019 MATHEMATICA max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* Jean-François 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 */ (PARI) x='x+O('x^33); Vec(serlaplace(exp(sin(x)))) \\ Joerg Arndt, Apr 01 2017 CROSSREFS a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|. Cf. A003724. - Franklin T. Adams-Watters, Apr 29 2010 Sequence in context: A280369 A280979 A281045 * A278292 A281520 A281634 Adjacent sequences:  A002014 A002015 A002016 * A002018 A002019 A002020 KEYWORD sign,easy,nice AUTHOR EXTENSIONS Extended with signs by Christian G. Bower, Nov 15 1998 STATUS approved

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Last modified October 14 16:48 EDT 2019. Contains 328022 sequences. (Running on oeis4.)