A005002 Number of rhyme schemes (see reference for precise definition). (Formerly M3465)

1, 4, 13, 41, 134, 471, 1819, 7778, 36703, 189381, 1057332, 6328261, 40300959, 271501240, 1925961025, 14332064197, 111528998198, 905134802555, 7643011810167, 67010181855706, 608890179868163, 5724496098183649

Offset

1,2

References

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 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

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J. Riordan, Cached copy of paper

Formula

a(k)=1. a(n) = k*a(n-1) + A000110(n-1) - A102661(n-1,k-2), k=2. - R. J. Mathar, Jul 15 2008

Maple

A000110 := proc(n) combinat[bell](n) ; end:
A005001:=n->if n = 0 then 0; else add(combinat[bell](k), k=0..n); fi;
A102661 := proc(n, k) add(combinat[stirling2](n, i), i=1..k) ; end:
beta := proc(n, k) if k= 1 then A005001(n) ; elif k= n then 1 ; else k*beta(n-1, k)+A000110(n-1)-A102661(n-1, k-2) ; fi ; end:
A005002 := proc(n) beta(n, 2) ; end:
seq(A005002(n), n=2..30) ; # R. J. Mathar, Jul 15 2008

Mathematica

a[1]=1; a[n_] := a[n] = 2a[n-1] + BellB[n]; a /@ Range[22]
(* Jean-François Alcover, May 19 2011, after R. J. Mathar *)
nxt[{n_, a_}]:={n+1, 2a+BellB[n+1]}; Transpose[NestList[nxt, {1, 1}, 30]] [[2]] (* Harvey P. Dale, Apr 20 2015 *)

Prog

(Haskell)
a005002 n = a005002_list !! (n-1)
a005002_list = 1 : zipWith (+) (map (* 2) a005002_list)
                               (drop 2 a000110_list)
-- Reinhard Zumkeller, Jun 19 2015

Crossrefs

Cf. A006505, A005003, A127021.

Cf. A000110.

Sequence in context: A239249 A141364 A001453 * A085507 A121654 A186202

Adjacent sequences: A004999 A005000 A005001 * A005003 A005004 A005005

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Jul 15 2008

Status

approved