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%I M2709 N1086 #62 Feb 13 2024 06:54:24
%S 1,1,1,0,-3,-8,-3,56,217,64,-2951,-12672,5973,309376,1237173,-2917888,
%T -52635599,-163782656,1126610929,12716052480,20058390573,
%U -495644917760,-3920482183827,4004259037184,256734635981833,1359174582304768
%N Expansion of e.g.f. exp(sin(x)).
%C 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
%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002017/b002017.txt">Table of n, a(n) for n=0..100</a>
%H E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F 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
%F 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
%F 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
%F 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
%F 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
%F 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
%e 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
%t max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* _Jean-François Alcover_, Jun 26 2013 *)
%o (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 */
%o (Maxima)
%o 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 */
%o (PARI) x='x+O('x^33); Vec(serlaplace(exp(sin(x)))) \\ _Joerg Arndt_, Apr 01 2017
%Y a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|.
%Y Cf. A003724. - _Franklin T. Adams-Watters_, Apr 29 2010
%K sign,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E Extended with signs by _Christian G. Bower_, Nov 15 1998
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