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A129247
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Invert transform of the Bell numbers.
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4
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1, 1, 3, 10, 36, 138, 560, 2402, 10898, 52392, 267394, 1450790, 8371220, 51327178, 333759746, 2295276480, 16639104002, 126718172670, 1010487248556, 8411744415418, 72899055533482, 656136245454232, 6120474697035762
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OFFSET
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0,3
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COMMENTS
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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."
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
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LINKS
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M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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FORMULA
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a(n) = Sum_{i=1..n} Bell(i)*a(n-i).
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
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
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
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EXAMPLE
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We have Bell(i) types of an integer i with i=1,2,...,n, where Bell(i) is the i-th Bell number.
We write i_j for integer i of type j.
a(2)=3 because of the 3 ordered arrangements
{1_1,1_1}
{2_1}, {2_2}.
a(3)=10 because of the 10 ordered arrangements
{1_1,1_1,1_1},
{1_1,2_1}, {2_1,1_1},
{1_1,2_2}, {2_2,1_1}
{3_1}, {3_2}, {3_3}, {3_4}, {3_5}.
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MAPLE
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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
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = Sum[BellB[i]*a[n - i], {i, 1, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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