cp's OEIS Frontend

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

%I A005003 M4416 #20 Aug 02 2015 20:30:32
%S A005003 1,7,35,156,670,2886,12797,59537,294585,1562324,8900568,54346140,
%T A005003 353937741,2444771767,17814457447,136308242144,1091001532590,
%U A005003 9105746802826,79041398643849,711994012088297,6642697774712213
%N A005003 Number of rhyme schemes (see reference for precise definition).
%D A005003 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.
%D A005003 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005003 Vincenzo Librandi, <a href="/A005003/b005003.txt">Table of n, a(n) for n = 1..200</a>
%H A005003 J. Riordan, <a href="/A005000/a005000.pdf">Cached copy of paper</a>
%F A005003 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
%p A005003 (Maple program from _R. J. Mathar_):
%p A005003 A000110 := proc(n) combinat[bell](n) ; end:
%p A005003 A102661 := proc(n,k) add(combinat[stirling2](n,i),i=1..k) ; end:
%p A005003 A005001:=n->if n = 0 then 0; else add(combinat[bell](k),k=0..n); fi;
%p A005003 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:
%p A005003 A005003 := proc(n) beta(n,3) ; end:
%p A005003 seq(A005003(n),n=3..30) ;
%t A005003 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_ *)
%Y A005003 Cf. A005002, A127021.
%K A005003 nonn
%O A005003 1,2
%A A005003 _N. J. A. Sloane_
%E A005003 More terms from _R. J. Mathar_, Jul 15 2008