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
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
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; 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]
T. Mansour and M. Shattuck, A statistic on n-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) 124(2) (2014), pp. 127-140.
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 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}.
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
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