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A035009 STIRLING transform of [1,1,2,4,8,16,32,...].

%I A035009 #90 Jul 05 2025 16:33:22
%S A035009 1,1,3,11,47,227,1215,7107,44959,305091,2206399,16913987,136823263,
%T A035009 1163490499,10366252031,96491364675,935976996127,9440144423875,
%U A035009 98800604237119,1071092025420867,12008090971866207,139014305916844739,1659578039401022079,20405708646650507075
%N A035009 STIRLING transform of [1,1,2,4,8,16,32,...].
%C A035009 Numerators of sequence that shifts left one place under 1/2 order binomial transform. (Denominators are 2^(n-1) for n > 0.) - _Franklin T. Adams-Watters_, Jul 31 2005
%C A035009 Row sums of triangle A137597 starting (1, 3, 11, 47, 227, ...). - _Gary W. Adamson_, Jan 29 2008
%C A035009 From _Gary W. Adamson_, Jul 22 2011: (Start)
%C A035009 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:
%C A035009   1,  1/2,  0,   0,   0,   0,  ...
%C A035009   1,   1,  1/2,  0,   0,   0,  ...
%C A035009   1,   2,   1,  1/2,  0,   0,  ...
%C A035009   1,   3,   3,   1,  1/2,  0,  ...
%C A035009   1,   4,   6,   4,   1,  1/2, ..., etc.
%C A035009 (End)
%C A035009 From _Bruno Berselli_, Mar 20 2013: (Start)
%C A035009 Note that, for t=A222391:
%C A035009 a(1)*t = Sum_{n >= 1} 1  /(Gamma(n/2)*Gamma((n+1)/2)),
%C A035009 a(2)*t = Sum_{n >= 1} n  /(Gamma(n/2)*Gamma((n+1)/2)),
%C A035009 a(3)*t = Sum_{n >= 1} n^2/(Gamma(n/2)*Gamma((n+1)/2)),
%C A035009 a(4)*t = Sum_{n >= 1} n^3/(Gamma(n/2)*Gamma((n+1)/2)),
%C A035009 a(5)*t = Sum_{n >= 1} n^4/(Gamma(n/2)*Gamma((n+1)/2)),
%C A035009 a(6)*t = Sum_{n >= 1} n^5/(Gamma(n/2)*Gamma((n+1)/2)), etc.
%C A035009 (End)
%C A035009 Except for the initial term, the main diagonal of A129340. - _Peter Bala_, Apr 14 2017
%H A035009 Seiichi Manyama, <a href="/A035009/b035009.txt">Table of n, a(n) for n = 0..558</a>
%H A035009 Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%H A035009 Mikhail Khovanov, Victor Ostrik and Yakov Kononov, <a href="https://arxiv.org/abs/2011.14758">Two-dimensional topological theories, rational functions and their tensor envelopes</a>, arXiv:2011.14758 [math.QA], 2020.
%H A035009 Toufik Mansour and Mark Shattuck, <a href="https://www.emis.de/journals/INTEGERS/papers/l67/l67.Abstract.html">A recurrence related to the Bell Numbers</a>, Integers 11 (2011), #A67.
%F A035009 a(n) = (1/2)*A001861(n), n > 0.
%F A035009 E.g.f.: (1 + exp(2*exp(x)-2))/2. - _Emeric Deutsch_, Feb 09 2002
%F A035009 a(n+1) = 1 + 2*Sum_{j=1..n} binomial(n, j)*a(j). - _Jon Perry_, Apr 25 2005
%F A035009 Define f_1(x), f_2(x), ... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x) = (d/dx)(x*f_n(x)). Then a(n) = e^(-2)*f_n(2). - _Milan Janjic_, May 30 2008
%F A035009 G.f.: 1 + x/(Q(0) - 2*x) where Q(k) =  1 - x*(k+1)/( 1 - 2*x/Q(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 22 2013
%F A035009 G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - x*(2*k+1)/(1 - x - 2*x/(1 - x*(2*k+2)/Q(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, May 13 2013
%F A035009 G.f.: 1 + Sum_{k>=1} 2^(k-1)*x^k/Product_{j=1..k} (1 - j*x). - _Ilya Gutkovskiy_, Jun 19 2018
%e A035009 Given the production matrix M, upper left term of M^5 = a(5)/2^4 = 227/16.
%p A035009 A035009 := proc(n) local a,b,i;
%p A035009 a := [seq(2,i=1..n-1)]; b := [seq(1,i=1..n-1)];
%p A035009 exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=2,%), 10+2*n)) end:
%p A035009 seq(A035009(n),n=0..19);  # _Peter Luschny_, Mar 30 2011
%p A035009 # second Maple program:
%p A035009 b:= proc(n, m) option remember;
%p A035009      `if`(n=0, ceil(2^(m-1)), m*b(n-1, m)+b(n-1, m+1))
%p A035009     end:
%p A035009 a:= n-> b(n, 0):
%p A035009 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 03 2021
%t A035009 1/(2*E^2)*Sum[(i + j)^n/(i!*j!), {i, 0, Infinity}, {j, 0, Infinity}] (* Starting from the 2nd term *) (* _Vladimir Reshetnikov_, Dec 31 2008 *)
%t A035009 Join[{1}, Table[BellB[n, 2]/2, {n, 1, 25}]] (* _Vaclav Kotesovec_, Jun 26 2022 *)
%o A035009 (PARI) x='x+O('x^99); Vec(serlaplace((1 + exp(2*exp(x)-2))/2)) \\ _Joerg Arndt_, Apr 01 2011
%Y A035009 Cf. A000110, A011782, A137597, A222391, A129340.
%K A035009 nonn,easy
%O A035009 0,3
%A A035009 _N. J. A. Sloane_