Search: id:A014182
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%I A014182
%S A014182 1,0,1,1,2,9,9,50,267,413,2180,17731,50533,110176,1966797,9938669,8638718,
%T A014182 278475061,2540956509,9816860358,27172288399,725503033401,5592543175252,
%U A014182 15823587507881,168392610536153,2848115497132448,20819319685262839
%V A014182 1,0,-1,1,2,-9,9,50,-267,413,2180,-17731,50533,110176,-1966797,9938669,
               -8638718,
%W A014182 -278475061,2540956509,-9816860358,-27172288399,725503033401,-5592543175252,
%X A014182 15823587507881,168392610536153,-2848115497132448,20819319685262839
%N A014182 Expansion of exp(1-x-exp(-x)).
%C A014182 E.g.f. A(x)=y satisfies (y+y'+y'')y-y'^2=0. - Michael Somos Mar 11 2004
%C A014182 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
%C A014182 Equals row sums of triangle A143987 and (shifted) = right border of A143987. 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008]
%C A014182 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2008: 
               (Start)
%C A014182 Equals the eigensequence of the inverse of Pascal's triangle, A007318.
%C A014182 Binomial transform shifts to the right: (1, 1, 0, -1, 1, 2, -9,...).
%C A014182 Double binomial transform = A109747 (End)
%C A014182 Convolved with A154107 = A000110, the Bell numbers. [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jan 04 2009]
%F A014182 E.g.f.: exp(1-x-exp(-x)).
%F A014182 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n+1,k+1). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 12 2006
%o A014182 (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
%o A014182 (PARI) {a(n)=if(n<0, 0, n!*polcoeff( exp( 1-x-exp( -x+x*O(x^n))), n))} 
               /* Michael Somos Mar 11 2004 */
%Y A014182 Essentially same as A000587. See also A014619.
%Y A014182 Cf. A025016.
%Y A014182 A143987 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008]
%Y A014182 A109747 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2008]
%Y A014182 A154107, A000110 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 
               2009]
%Y A014182 Sequence in context: A003678 A109322 A000587 this_sequence A131463 A065644 
               A043065
%Y A014182 Adjacent sequences: A014179 A014180 A014181 this_sequence A014183 A014184 
               A014185
%K A014182 sign,easy,nice
%O A014182 0,5
%A A014182 Noam Elkies (elkies(AT)math.harvard.edu)

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