login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005001 a(0) = 0; for n>0, a(n) = Sum_k={0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.
(Formerly M1194)
17
0, 1, 2, 4, 9, 24, 76, 279, 1156, 5296, 26443, 142418, 820988, 5034585, 32679022, 223578344, 1606536889, 12086679036, 94951548840, 777028354999, 6609770560056, 58333928795428, 533203744952179, 5039919483399502, 49191925338483848, 495150794633289137 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Counts rhyme schemes.

Row sums of triangle A137596 starting with offset 1. - Gary W. Adamson, Jan 29 2008

With offset 1 = binomial transform of the Bell numbers, A000110 starting (1, 1, 1, 2, 5, 15, 52, 203,...). - Gary W. Adamson, Dec 04 2008

a(n) is the number of partitions of the set {1,2,...,n} in which n is either a singleton or it is in a block of consecutive integers. Example: a(3)=4 because we have 123, 1-23, 12-3, and 1-2-3. Deleting the blocks containing n=3, we obtain: empty, 1, 12, 1-2, i.e. all the partitions of the sets: empty, {1}, and {1,2}. - Emeric Deutsch, May 01 2010

REFERENCES

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

J. Riordan, Cached copy of paper

FORMULA

a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).

Sum_{i=1..n} Bell(i) = 1 + C(n, 2) + 2*C(n-3, 1) + 8*C(n-4, 1) + C(n-3, 2) + 22*C(n-5, 1) + 13*C(n-4, 2) + 52*C(n-6, 1) + 74*C(n-5, 2) + 10*C(n-4, 3) + 114*C(n-7, 1) + 314*C(n-6, 2) + 134*C(n-5, 3) + 3*C(n-4, 4) + 240*C(n-8, 1) + 1155*C(n-7, 2) + 1024*C(n-6, 3) + 134*C(n-5, 4) + 494*C(n-9, 1) + ..... . - André F. Labossière, Feb 11 2005

a(n) = A000110(n) - A171859(n). [Emeric Deutsch, May 01 2010]

G.f.: x*( 1 + (G(0)+1)*x/(1-x) ) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 20 2012

G.f.: x*G(0)/(1-x^2) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012

G.f.: x*( G(0) - 1 )/(1-x) where G(k) =  1 + (1-x)/(1-x*k)/(1-x/(x+(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2013

G.f.: (G(0)-1)*x/(1-x^2) where G(k) = 1 + 1/(1-k*x)/(1-x/(x+1/G(k+1) );  (continued fraction). - Sergei N. Gladkovskii, Feb 06 2013

G.f.: x/(1-x)/(1-x*Q(0)), where Q(k)= 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013

E.g.f. A(x) satisfies: A'(x) = A(x) + exp(exp(x)-1). - Geoffrey Critzer, Feb 04 2014

MAPLE

with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # Emeric Deutsch, May 01 2010

MATHEMATICA

nn=20; Range[0, nn]!CoefficientList[Series[Exp[-1](-Exp[Exp[x]]+Exp[1+x]-Exp[x]ExpIntegralEi[1]+Exp[x]ExpIntegralEi[Exp[x]]), {x, 0, nn}], x] (* Geoffrey Critzer, Feb 04 2014 *)

CROSSREFS

Partial sums of A000110, partial sums give A029761.

Equals A024716(n-1) + 1.

Cf. A102735, A094262, A000110, A008277, A102639, A003422, A000166, A000204, A000045, A000108.

Cf. A137596.

Cf. A171859. - Emeric Deutsch, May 01 2010

Sequence in context: A236756 A125654 A141824 * A091151 A093542 A000667

Adjacent sequences:  A004998 A004999 A005000 * A005002 A005003 A005004

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 22 01:35 EDT 2014. Contains 245921 sequences.