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%I #47 Feb 15 2022 07:32:58
%S 1,1,3,10,36,138,560,2402,10898,52392,267394,1450790,8371220,51327178,
%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 appears in [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 Equals INVERT transform of the Bell sequence starting with offset 1: (1, 2, 5, ...), while A137551 = INVERT transform of the Bell sequence starting with offset 0: (1, 1, 2, 5, 15, 52, ...). - _Gary W. Adamson_, May 24 2009
%H Alois P. Heinz, <a href="/A129247/b129247.txt">Table of n, a(n) for n = 0..576</a>
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H T. Mansour and M. Shattuck, <a href="https://doi.org/10.1007/s12044-014-0166-7">A statistic on n-color compositions and related sequences</a>, Proc. Indian Acad. Sci. (Math. Sci.) 124(2) (2014), pp. 127-140.
%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 of the 3 ordered arrangements
%e {1_1,1_1}
%e {2_1}, {2_2}.
%e a(3)=10 because of the 10 ordered arrangements
%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 0 to 40 do printf("%d,",A129247(n)) ; od: # _R. J. Mathar_, Aug 25 2008
%t a[0] = 1; a[n_] := a[n] = Sum[BellB[i]*a[n - i], {i, 1, n}];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 09 2017 *)
%Y Cf. A000110, A055887, A083355.
%K nonn
%O 0,3
%A _Thomas Wieder_, May 10 2008
%E Extended by _R. J. Mathar_, Aug 25 2008
%E a(0)=1 prepended by _Alois P. Heinz_, Sep 22 2017
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