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A005011
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Shifts one place left under 5th order binomial transform.
(Formerly M4240)
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5
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1, 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, 11458179765541, 249255304141006, 5725640423174901, 138407987170952351, 3510263847256823056
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+5 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=5, otherwise F(k+1)=F(k); see examples in A004211, A004212, and A004213, and Fxtbook link. [Joerg Arndt, Apr 30 2011]
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REFERENCES
| A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..66
Joerg Arndt, Fxtbook, section 17.3.5, pp. 366-368
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
N. J. A. Sloane, Transforms
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FORMULA
| a(n)=sum((5^(n-m))*stirling2(n, m), m=0..n), n>=0.
E.g.f.: exp((exp(5*x)-1)/5).
O.g.f. A(x) satisfies A'(x)/A(x) = exp(5*x).
E.g.f.: exp(int(t=0..x, exp(5*t))). [Joerg Arndt, Apr 30 2011]
O.g.f.: sum(k>=0, x^k/prod(j=1..k, (1-5*j*x))). [Joerg Arndt, Apr 30 2011]
Define f_1(x),f_2(x),... such that f_1(x)=e^x, f_{n+1}(x)=diff(x*f_n(x),x), for n=2,3,.... Then a(n)=e^{-1/5}*5^{n-1}*f_n(1/5). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008
a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (5,5,5,...) is appended to the right of Pascal's triangle:
1, 5, 0, 0, 0,...
1, 1, 5, 0, 0,...
1, 2, 1, 5, 0,...
1, 3, 3, 1, 5,...
... - Gary W. Adamson, Jul 29 2011
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PROG
| (Pari) x='x+O('x^66); /* that many terms */
egf=exp(intformal(exp(5*x))); /* = 1 + x + 3*x^2 + 41/6*x^3 + 331/24*x^4 + ... */
/* egf=exp(1/5*(exp(5*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */
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CROSSREFS
| Cf. A075500 (row sums).
A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A004213 (s(k)<=F(k)+4), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]
Sequence in context: A095177 A199553 A083430 * A009122 A184140 A094869
Adjacent sequences: A005008 A005009 A005010 * A005012 A005013 A005014
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KEYWORD
| easy,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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