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A004213
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Shifts one place left under 4th order binomial transform.
(Formerly M3956)
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5
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1, 1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, 1625661357673, 29905322979421, 580513190237573, 11850869542405409, 253669139947767777, 5678266212792053029
(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)+4 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=4, otherwise F(k+1)=F(k); see example 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((4^(n-m))*stirling2(n, m), m=0..n), n>=0.
E.g.f.: exp((exp(4*x)-1)/4).
O.g.f. A(x) satisfies A'(x)/A(x) = e^(4x).
E.g.f.: exp(int(t=0..x, exp(4*t))). [Joerg Arndt, Apr 30 2011]
O.g.f.: sum(k>=0, x^k/prod(j=1..k, (1-4*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^{-1/4}*4^{n-1}*f_n(1/4). - 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 (4,4,4,...) is appended to the right of Pascal's triangle:
1, 4, 0, 0, 0,...
1, 1, 4, 0, 0,...
1, 2, 1, 4, 0,...
1, 3, 3, 1, 4,...
... - Gary W. Adamson, Jul 29 2011
G.f. satisfies A(x)=1+x/(1-4*x)*A(4*x/(1-4*x)). a(n)=sum(4^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n), n>0, a(0)=1. [ From Vladimir Kruchinin, Nov 28 2011]
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EXAMPLE
| Restricted growth strings: a(0)=1 corresponds to the empty string, a(1)=1 to [0],
a(2)=3 to [00], [01], [02], [03], and [04], a(3) = 29 to
RGS F
1: [ 0 0 0 ] [ 0 0 0 ]
2: [ 0 0 1 ] [ 0 0 0 ]
3: [ 0 0 2 ] [ 0 0 0 ]
4: [ 0 0 3 ] [ 0 0 0 ]
5: [ 0 0 4 ] [ 0 0 4 ]
6: [ 0 1 0 ] [ 0 0 0 ]
7: [ 0 1 1 ] [ 0 0 0 ]
8: [ 0 1 2 ] [ 0 0 0 ]
9: [ 0 1 3 ] [ 0 0 0 ]
10: [ 0 1 4 ] [ 0 0 4 ]
11: [ 0 2 0 ] [ 0 0 0 ]
12: [ 0 2 1 ] [ 0 0 0 ]
13: [ 0 2 2 ] [ 0 0 0 ]
14: [ 0 2 3 ] [ 0 0 0 ]
15: [ 0 2 4 ] [ 0 0 4 ]
16: [ 0 3 0 ] [ 0 0 0 ]
17: [ 0 3 1 ] [ 0 0 0 ]
18: [ 0 3 2 ] [ 0 0 0 ]
19: [ 0 3 3 ] [ 0 0 0 ]
20: [ 0 3 4 ] [ 0 0 4 ]
21: [ 0 4 0 ] [ 0 4 4 ]
22: [ 0 4 1 ] [ 0 4 4 ]
23: [ 0 4 2 ] [ 0 4 4 ]
24: [ 0 4 3 ] [ 0 4 4 ]
25: [ 0 4 4 ] [ 0 4 4 ]
26: [ 0 4 5 ] [ 0 4 4 ]
27: [ 0 4 6 ] [ 0 4 4 ]
28: [ 0 4 7 ] [ 0 4 4 ]
29: [ 0 4 8 ] [ 0 4 8 ]
[Joerg Arndt, Apr 30 2011]
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PROG
| (Pari) x='x+O('x^66); /* that many terms */
egf=exp(intformal(exp(4*x))); /* = 1 + x + 5/2*x^2 + 29/6*x^3 + 67/8*x^4 + ... */
/* egf=exp(1/4*(exp(4*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */
(Maxima)
a(n):=if n=0 then 1 else sum(4^(n-k)*binomial(n-1, k-1)*a(k-1), k, 1, n); [From Vladimir Kruchinin, Nov 28 2011]
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CROSSREFS
| Cf. A075499 (row sums).
A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]
Sequence in context: A094710 A108453 A201203 * A105277 A103213 A057588
Adjacent sequences: A004210 A004211 A004212 * A004214 A004215 A004216
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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