A005003 Number of rhyme schemes (see reference for precise definition). (Formerly M4416)

1, 7, 35, 156, 670, 2886, 12797, 59537, 294585, 1562324, 8900568, 54346140, 353937741, 2444771767, 17814457447, 136308242144, 1091001532590, 9105746802826, 79041398643849, 711994012088297, 6642697774712213

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=3. - R. J. Mathar, Jul 15 2008

Maple

(Maple program from R. J. Mathar):
A000110 := proc(n) combinat[bell](n) ; end:
A102661 := proc(n, k) add(combinat[stirling2](n, i), i=1..k) ; end:
A005001:=n->if n = 0 then 0; else add(combinat[bell](k), k=0..n); fi;
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:
A005003 := proc(n) beta(n, 3) ; end:
seq(A005003(n), n=3..30) ;

Mathematica

a[1] = 1; a[n_] := 3*a[n-1] + BellB[n+1] - StirlingS2[n+1, 1]; a /@ Range[21] (* Jean-François Alcover, May 20 2011, after R. J. Mathar *)

Crossrefs

Cf. A005002, A127021.

Sequence in context: A006095 A171477 A265612 * A243382 A242577 A163348

Adjacent sequences: A005000 A005001 A005002 * A005004 A005005 A005006

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved