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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Crossrefs

Cf. A006505, A005003, A127021.

Cf. A000110.

Programs

Haskell

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

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 *)

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

Extensions

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