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A035009
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STIRLING transform of [1,1,2,4,8,16,32, ...].
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10
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1, 1, 3, 11, 47, 227, 1215, 7107, 44959, 305091, 2206399, 16913987, 136823263, 1163490499, 10366252031, 96491364675, 935976996127, 9440144423875, 98800604237119, 1071092025420867, 12008090971866207
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Numerators of sequence that shifts left one place under 1/2 order binomial transform. (Denominators are 2^(n-1) for n>0.) - Frank Adams-Watters, Jul 31 2005
Row sums of triangle A137597 starting (1, 3, 11, 47, 227,...). - Gary W. Adamson, Jan 29 2008
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REFERENCES
| Toufik Mansour and Mark Shattuck, A RECURRENCE RELATED TO THE BELL NUMBERS, INTEGERS 11 (2011), #A67; http://www.emis.de/journals/INTEGERS/papers/l67/l67.pdf.
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FORMULA
| E.g.f.: [1 + exp(2*exp(x)-2)]/2. - Emeric Deutsch, Feb 09, 2002
a(n+1) = 1 + 2*sum { j=1, n, binomial(n, j)*a(j) } - Jon Perry, Apr 25 2005
Define f_1(x),f_2(x),... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x)=diff(x*f_n(x),x). Then a(n)=e^{-2}*f_n(2). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n)/2^(n-1) = upper left term in M^n, M = an infinite square production matrix in which a column of (1/2, 1/2, 1/2,...) is appended to the right of Pascal's triangle, as follows:
1, 1/2, 0, 0, 0, 0,...
1, 1, 1/2, 0, 0, 0,...
1, 2, 1, 1/2, 0, 0,...
1, 3, 3, 1, 1/2, 0,...
1, 4, 6, 4, 1, 1/2,...
... (End)
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EXAMPLE
| Given the production matrix M, upper left term of M^5 = a(5)/2^4 = 227/16
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MAPLE
| A035009 := proc(n) local a, b, i;
a := [seq(2, i=1..n-1)]; b := [seq(1, i=1..n-1)];
exp(-x)*hypergeom(a, b, x); round(evalf(subs(x=2, %), 10+2*n)) end:
seq(A035009(n), n=0..19); # - Peter Luschny, Mar 30 2011
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MATHEMATICA
| 1/(2*E^2)*Sum[(i + j)^n/(i!*j!), {i, 0, Infinity}, {j, 0, Infinity}] (* Starting from the 2nd term *) [From Vladimir Reshetnikov, Dec 31 2008]
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PROG
| (PARI) x='x+O('x^99); /* that many terms */
Vec(serlaplace((1 + exp(2*exp(x)-2))/2)) /* show terms */ /* Joerg Arndt, Apr 1 2011 */
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CROSSREFS
| Equals (1/2) A001861(n), n>0.
Cf. A000110.
Cf. A137597.
Sequence in context: A174347 A062146 A090365 * A051296 A030832 A030865
Adjacent sequences: A035006 A035007 A035008 * A035010 A035011 A035012
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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