A052872 A simple grammar.

1, 1, 3, 9, 32, 117, 460, 1850, 7680, 32467, 139631, 608280, 2679881, 11916492, 53417407, 241118245, 1095030281, 4999847240, 22938576846, 105690791104, 488860041249, 2269080989966, 10565660913109, 49340702712984, 231031730790560

Offset

0,3

Links

Table of n, a(n) for n=0..24.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 843

Formula

G.f. A(x) satisfies [from Paul D. Hanna, Oct 16 2013]:

(1) A(x) = exp( Sum_{n>=1} x^n/n * A(x^n)/(1 - x^n*A(x^n)) );

(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*A(x^(n/d))^d ).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 117*x^5 + 460*x^6 +...
where
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2/2*A(x^2)/(1-x^2*A(x^2)) + x^3/3*A(x^3)/(1-x^3*A(x^3)) + x^4/4*A(x^4)/(1-x^4*A(x^4)) + x^5/5*A(x^5)/(1-x^5*A(x^5)) + x^6/6*A(x^6)/(1-x^6*A(x^6)) +... - Paul D. Hanna, Oct 16 2013

Maple

spec := [S, {C=Sequence(B, 1 <= card), S=Set(C), B=Prod(Z, S)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(k=1, n, x^k/k*subst(A, x, x^k)/(1-x^k*subst(A, x, x^k+x*O(x^n)) )))); polcoeff(A, n)} \\ Paul D. Hanna, Oct 16 2013
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(k=1, n, x^k/k*sumdiv(k, d, d*subst(A, x, x^(k/d) +x*O(x^n))^d)))); polcoeff(A, n)} \\ Paul D. Hanna, Oct 16 2013

Crossrefs

Sequence in context: A194353 A148985 A148986 * A122452 A192206 A091841

Adjacent sequences: A052869 A052870 A052871 * A052873 A052874 A052875

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved