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

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

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.yu) 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: A047091 A121092 A030981 this_sequence A032265 A110143 A074410

Adjacent sequences: A000293 A000294 A000295 this_sequence A000297 A000298 A000299

nonn,easy

njas

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

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