A193040 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A131507(n), where A131507 is defined as "2*n+1 appears n times.".

1, 1, 2, 7, 29, 129, 600, 2889, 14293, 72228, 371208, 1934236, 10194853, 54258010, 291175463, 1573878211, 8560931357, 46825444031, 257386132988, 1421034475176, 7876770462043, 43817869686744, 244552276036950, 1368945007588648, 7683977372121530

Offset

0,3

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..24.

Formula

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 600*x^6 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^5 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^7 +...+ x^n*A(-x)^A131507(n) +...
where A131507 begins: [1,3,3,5,5,5,7,7,7,7,9,9,9,9,9,11,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^2)*A(-x)^3 + x^3*(1-x^3)*A(-x)^5 + x^6*(1-x^4)*A(-x)^7 + x^10*(1-x^5)*A(-x)^9 +...

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)^(2*floor(sqrt(2*m)+1/2)-1) ), #A)); if(n<0, 0, A[n+1])}

Crossrefs

Cf. A193039, A193037, A192455, A193050, A131507.

Sequence in context: A054321 A344502 A150664 * A200755 A132262 A371431

Adjacent sequences: A193037 A193038 A193039 * A193041 A193042 A193043

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2011

Status

approved