0,5

a(n+2)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=A000110(n) for k=0,1,...,n. - Michael Somos, Oct 07 2003

Number of complete rhyming schemes.

Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the _central_ moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005

Row sums of the triangle of associated Stirling numbers of second kind A008299 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 10 2005

a(n) = number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

E. Bach, Random bisection and evolutionary walks, J. Applied Probability, v. 38, pp. 582-596, 2001.

H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 701-702.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

Martin Gardner in Sci. Amer. May 1977.

G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 228.

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.

T. D. Noe, Table of n, a(n) for n=0..100

S. R. Finch, Moments of sums

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 16

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

J. Riordan, Cached copy of paper

Index entries for related partition-counting sequences

E.g.f.: exp(exp(x) - 1 - x).

B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker]

Inverse binomial transform of Bell numbers (A000110).

a(n)= sum((k)^n/(k+1)!, k = -1 .. infinity)/exp(1). - Vladeta Jovovic (vladeta(AT)eunet.rs) and Karol A. PENSON (penson(AT)lptl.jussieu.fr), Feb 02 2003

a(n)= sum(((-1)^(n-k))*binomial(n, k)*Bell(k), k=0..n) = (-1)^n + Bell(n) - A087650(n), with Bell(n)=A000110(n). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003

O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 17 2006

a(n)= sum(k=0..n) {(-1)^(n-k)* sum(j=0..k)[(-1)^j * binomial(k,j)* (1-j)^n]/ k!} = sum over row n of A105794 - Tom Copeland (tcjpn(AT)msn.com), Jun 05 2006

a(n)=(-1)^n + sum[(-1)^(j-1)*B(n-j),j=1..n], where B(q) are the Bell numbers (A000110). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2006

spec := [ B, {B=Set(Set(Z, card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];

with(combinat): a:=n->(-1)^n+sum((-1)^(j-1)*bell(n-j), j=1..n): seq(a(n), n=0..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2006

f:=exp(exp(x)-1-x): fser:=series(f, x=0, 31): 1, seq(n!*coeff(fser, x^n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 22 2006

G:={P=Set(Set(Atom, card>=2))}:combstruct[gfsolve](G, unlabeled, x):seq(combstruct[count]([P, G, labeled], size=i), i=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007

(PARI) a(n)=if(n<2, n==0, subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n)/x)-1)))), x, n))

Cf. A000110, A006505, A057814, A057837.

Cf. A105794.

Sequence in context: A151455 A149269 A149270 this_sequence A032265 A151273 A149271

Adjacent sequences: A000293 A000294 A000295 this_sequence A000297 A000298 A000299

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms, new description from Christian G. Bower (bowerc(AT)usa.net), Nov 15 1999.

Becker reference from D. E. Knuth, Dec 20 2003