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A004211
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Shifts one place left under 2nd order binomial transform.
(Formerly M2900)
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13
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1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, 5120905441, 56878092067, 664920021819, 8155340557697, 104652541401025, 1401572711758403, 19546873773314571, 283314887789276721
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Equals the eigensequence of A038207, the square of Pascal's triangle. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
Contribution from Paul Barry (pbarry(AT)wit.ie), Dec 04 2009: (Start)
Note that the g.f. of the second binomial transform is
1/(1-2x-x/(1-2x/(1-2x-x/(1-4x/(1-2x-x/(1-6x/(1-2x-x/(1-8x/(1-... (continued fraction). (End)
Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+2 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=2, otherwise F(k+1)=F(k); see example and Fxtbook link. [Joerg Arndt, Apr 30 2011]
It appears that the infinite set of "Shifts 1 place left under N-th order binomial transform" sequences has a production matrix of:
1, N, 0, 0, 0,...
1, 1, N, 0, 0,...
1, 2, 1, N, 0,...
1, 3, 3, 1, N,...
...(where a diagonal of (N,N,N,...) is appended to the right of Pascal's triangle). a(n) in each sequence is the upper left term of M^n such that N=1 generates A000110, then (N=2 - A004111), (N=3 - A004212), (N=4 - A004213), (N=5 - A005011),... - Gary W. Adamson, Jul 29 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
N. J. A. Sloane, Transforms
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FORMULA
| E.g.f.: exp(sinh(x)*exp(x)).
O.g.f. A(x) satisfies A'(x)/A(x) = e^(2x).
E.g.f.: exp(int(t=0..x, exp(2*t))) = exp(1/2*(exp(2*x)-1)). [Joerg Arndt, Apr 30 2011, and May 13 2011]
a_n=sum(2^(n-k)*stirling2(n, k), k=0..n). - Emeric Deutsch, Feb 11 2002
G.f.: sum{k>=0, x^k/prod[l=1..k, 1-2*l*x]}. - R. Stephan, Apr 18 2004
Stirling transform of A000085. - Vladeta Jovovic (vladeta(AT)eunet.rs), May 14 2004
O.g.f.: A(x) = 1/(1-x-2*x^2/(1-3*x-4*x^2/(1-5*x-6*x^2/(1-... -(2*n-1)*x-2*n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 17 2006
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^{-1/2}*2^{n-1}*f_n(1/2). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008
G.f.: 1/(1-x/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-x/(1-8x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Dec 04 2009]
a(n) = upper left term in M^n, M = an infinite square production matrix with an appended diagonal of (2,2,2,...) to the right of Pascal's triangle:
1, 2, 0, 0, 0,...
1, 1, 2, 0, 0,...
1, 2, 1, 2, 0,...
1, 3, 3, 1, 2,...
... - Gary W. Adamson, Jul 29 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A000110. - Peter Bala, Nov 25 2011
G.f. A(x) satisfies A(x)=1+x/(1-2*x)*A(2*x/(1-2*x)), a(n)=sum(k=1..n, binomial(n-1,k-1)*2^(n-k)*a(k-1), 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], and [02]; a(3) = 11 to
RGS F
1: [ 0 0 0 ] [ 0 0 0 ]
2: [ 0 0 1 ] [ 0 0 0 ]
3: [ 0 0 2 ] [ 0 0 2 ]
4: [ 0 1 0 ] [ 0 0 0 ]
5: [ 0 1 1 ] [ 0 0 0 ]
6: [ 0 1 2 ] [ 0 0 2 ]
7: [ 0 2 0 ] [ 0 2 2 ]
8: [ 0 2 1 ] [ 0 2 2 ]
9: [ 0 2 2 ] [ 0 2 2 ]
10: [ 0 2 3 ] [ 0 2 2 ]
11: [ 0 2 4 ] [ 0 2 4 ]
[Joerg Arndt, Apr 30 2011]
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MATHEMATICA
| Table[ Sum[ StirlingS2[ n, k ] 2^(-k+n), {k, n} ], {n, 16} ] (Wouter L. J. MEEUSSEN)
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PROG
| (Pari) x='x+O('x^66); /* that many terms */
egf=exp(intformal(exp(2*x))); /* = 1 + x + 3/2*x^2 + 11/6*x^3 + 49/24*x^4 + ... */
/* egf=exp(1/2*(exp(2*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */
(Maxima)
a(n):=if n=0 then 1 else sum(2^(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. A075497 (row sums).
A038207 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
A004212 (RGS where s(k)<=F(k)+3), A004213 (s(k)<=F(k)+4), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]
Sequence in context: A025539 A172440 A074528 * A180869 A001339 A012316
Adjacent sequences: A004208 A004209 A004210 * A004212 A004213 A004214
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KEYWORD
| nonn,easy,nice,eigen
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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