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 A129247 Invert transform of the Bell numbers. 2

%I

%S 1,3,10,36,138,560,2402,10898,52392,267394,1450790,8371220,51327178,

%T 333759746,2295276480,16639104002,126718172670,1010487248556,

%U 8411744415418,72899055533482,656136245454232,6120474697035762

%N Invert transform of the Bell numbers.

%C The following definition of the invert transform leads to the understanding of A129247 [M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and its Applications, 226-228 (1995), 57-72]: "b_n is the number of ordered arrangements of postage stamps of total value n that can be formed if we have a_i types of stamps of value i, i >= 1."

%C Hankel transform is A000178. [_Paul Barry_, Jan 08 2009]

%C A129247 = INVERT transform of the Bell sequence starting with offset 1: (1, 2, 5,...). A137551 = INVERT transform of the Bell sequence starting with offset 0: (1, 1, 2, 5, 15, 52,...). [_Gary W. Adamson_, May 24 2009]

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = Sum_{i=1..n} Bell(i)*a(n-i).

%F G.f.: 1/(U(0) - 2*x) where U(k) = 1 - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Nov 12 2012

%F G.f.: 1/( Q(0) - 2*x ) where Q(k)=1 + x/(x*k - 1 )/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Feb 23 2013

%F G.f.: 1/(Q(0) - x), where Q(k)= 1 - x - x/(1 - x*(2*k+1)/(1 - x - x/(1 - 2*x*(k+1)/Q(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, May 12 2013

%e We have Bell(i) types of an integer i with i=1,2,...,n, where Bell(i) is the i-th Bell number.

%e We write i_j for integer i of type j.

%e a(2)=3 because

%e {1_1,1_1}

%e {2_1}, {2_2}.

%e a(3)=10 because

%e {1_1,1_1,1_1},

%e {1_1,2_1}, {2_1,1_1},

%e {1_1,2_2}, {2_2,1_1}

%e {3_1}, {3_2}, {3_3}, {3_4}, {3_5}.

%p A129247 := proc(n) option remember ; local i ; if n <= 1 then 1 ; else add(combinat[bell](i)*procname(n-i),i=1..n) ; fi ; end: for n from 1 to 40 do printf("%d,",A129247(n)) ; od: # _R. J. Mathar_, Aug 25 2008

%Y Cf. A000110, A055887, A083355.

%K nonn,changed

%O 1,2

%A _Thomas Wieder_, May 10 2008

%E Extended by _R. J. Mathar_, Aug 25 2008

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