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A014182
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Expansion of exp(1-x-exp(-x)).
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9
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1, 0, -1, 1, 2, -9, 9, 50, -267, 413, 2180, -17731, 50533, 110176, -1966797, 9938669, -8638718, -278475061, 2540956509, -9816860358, -27172288399, 725503033401, -5592543175252, 15823587507881, 168392610536153, -2848115497132448, 20819319685262839
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| E.g.f. A(x)=y satisfies (y+y'+y'')y-y'^2=0. - Michael Somos Mar 11 2004
The 10-adic sum: B(n) = Sum_{k>=0} k^n*k! simplifies to: B(n) = A014182(n)*B(0) + A014619(n) for n>=0, where B(0) is the 10-adic sum of factorials (A025016); a result independent of base. - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 12 2006
Equals row sums of triangle A143987 and (shifted) = right border of A143987. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2008: (Start)
Equals the eigensequence of the inverse of Pascal's triangle, A007318.
Binomial transform shifts to the right: (1, 1, 0, -1, 1, 2, -9,...).
Double binomial transform = A109747 (End)
Convolved with A154107 = A000110, the Bell numbers. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 2009]
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FORMULA
| E.g.f.: exp(1-x-exp(-x)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n+1,k+1). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 12 2006
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MATHEMATICA
| With[{nn=30}, CoefficientList[Series[Exp[1-x-Exp[-x]], {x, 0, nn}], x] Range[0, nn]!] (* From Harvey P. Dale, Jan 15 2012 *)
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PROG
| (PARI) {a(n)=sum(j=0, n, (-1)^(n-j)*Stirling2(n+1, j+1))} /* Stirling2 defined by: */ {Stirling2(n, k)=(1/k!)*sum(i=0, k, (-1)^(k-i)*binomial(k, i)*i^n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 12 2006
(PARI) {a(n)=if(n<0, 0, n!*polcoeff( exp( 1-x-exp( -x+x*O(x^n))), n))} /* Michael Somos Mar 11 2004 */
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CROSSREFS
| Essentially same as A000587. See also A014619.
Cf. A025016.
A143987 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008]
A109747 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2008]
A154107, A000110 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 2009]
Sequence in context: A168333 A109322 A000587 * A131463 A065644 A043065
Adjacent sequences: A014179 A014180 A014181 * A014183 A014184 A014185
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KEYWORD
| sign,easy,nice
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AUTHOR
| Noam Elkies (elkies(AT)math.harvard.edu)
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