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A002017
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Expansion of e.g.f. exp(sin(x)).
(Formerly M2709 N1086)
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20
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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
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* Jean-François Alcover, Jun 26 2013 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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