A193039 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A002024(n+1), where A002024 is defined as "n appears n times.".

1, 1, 1, 2, 5, 13, 34, 91, 251, 709, 2035, 5913, 17366, 51483, 153858, 463001, 1401751, 4266619, 13048709, 40078032, 123570957, 382331356, 1186699353, 3694028136, 11529606672, 36073811897, 113123222246, 355485228001, 1119275386080, 3530531671842

Offset

0,4

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

Links

Table of n, a(n) for n=0..29.

Formula

G.f. satisfies: 1-x = Sum_{n>=1} x^(n*(n-1)/2) * (1-x^n) * A(-x)^n.

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 34*x^6 + 91*x^7 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^3 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^4 +...+ x^n*A(-x)^A002024(n+1) +...
where A002024 begins: [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^2)*A(-x)^2 + x^3*(1-x^3)*A(-x)^3 + x^6*(1-x^4)*A(-x)^4 + x^10*(1-x^5)*A(-x)^5 +...

Prog

(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^floor(sqrt(2*m)+1/2) ), #A)); if(n<0, 0, A[n+1])}

Crossrefs

Cf. A193040, A193037, A192455, A193050, A002024.

Sequence in context: A090827 A109192 A192313 * A062465 A261237 A318229

Adjacent sequences: A193036 A193037 A193038 * A193040 A193041 A193042

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2011

Status

approved