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 A005002 Number of rhyme schemes (see reference for precise definition). (Formerly M3465) 3

%I M3465

%S 1,4,13,41,134,471,1819,7778,36703,189381,1057332,6328261,40300959,

%T 271501240,1925961025,14332064197,111528998198,905134802555,

%U 7643011810167,67010181855706,608890179868163,5724496098183649

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

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

%H Vincenzo Librandi, <a href="/A005002/b005002.txt">Table of n, a(n) for n = 1..200</a>

%H J. Riordan, <a href="/A005000/a005000.pdf">Cached copy of paper</a>

%F 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

%p A000110 := proc(n) combinat[bell](n) ; end:

%p A005001:=n->if n = 0 then 0; else add(combinat[bell](k),k=0..n); fi;

%p A102661 := proc(n,k) add(combinat[stirling2](n,i),i=1..k) ; end:

%p 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 A005002 := proc(n) beta(n,2) ; end:

%p seq(A005002(n),n=2..30) ; # _R. J. Mathar_, Jul 15 2008

%t a[1]=1; a[n_] := a[n] = 2a[n-1] + BellB[n]; a /@ Range[22]

%t (* _Jean-François Alcover_, May 19 2011, after _R. J. Mathar_ *)

%t nxt[{n_,a_}]:={n+1,2a+BellB[n+1]}; Transpose[NestList[nxt,{1,1},30]] [[2]] (* _Harvey P. Dale_, Apr 20 2015 *)

%o a005002 n = a005002_list !! (n-1)

%o a005002_list = 1 : zipWith (+) (map (* 2) a005002_list)

%o (drop 2 a000110_list)

%o -- _Reinhard Zumkeller_, Jun 19 2015

%Y Cf. A005000, A005003, A127021.

%Y Cf. A000110.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _R. J. Mathar_, Jul 15 2008

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