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A129247 Invert transform of the Bell numbers. 2
1, 1, 3, 10, 36, 138, 560, 2402, 10898, 52392, 267394, 1450790, 8371220, 51327178, 333759746, 2295276480, 16639104002, 126718172670, 1010487248556, 8411744415418, 72899055533482, 656136245454232, 6120474697035762 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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."

Hankel transform is A000178. [Paul Barry, Jan 08 2009]

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]

REFERENCES

T. Mansour, M. Shattuck, A statistic on n-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 127-140.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..576

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 arXiv version]

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]

N. J. A. Sloane, Transforms

FORMULA

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

EXAMPLE

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

{1_1,1_1}

{2_1}, {2_2}.

a(3)=10 because

{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}.

MAPLE

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

MATHEMATICA

a[0] = 1; a[n_] := a[n] = Sum[BellB[i]*a[n - i], {i, 1, n}];

Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 09 2017 *)

CROSSREFS

Cf. A000110, A055887, A083355.

Sequence in context: A149041 A307346 A202834 * A162162 A149042 A081921

Adjacent sequences:  A129244 A129245 A129246 * A129248 A129249 A129250

KEYWORD

nonn

AUTHOR

Thomas Wieder, May 10 2008

EXTENSIONS

Extended by R. J. Mathar, Aug 25 2008

a(0)=1 prepended by Alois P. Heinz, Sep 22 2017

STATUS

approved

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Last modified June 19 17:16 EDT 2019. Contains 324222 sequences. (Running on oeis4.)