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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.". 6
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 (list; graph; refs; listen; history; text; internal format)
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: A150663 A054321 A150664 * A200755 A132262 A007852

Adjacent sequences:  A193037 A193038 A193039 * A193041 A193042 A193043

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 14 2011

STATUS

approved

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Last modified February 25 18:49 EST 2018. Contains 299655 sequences. (Running on oeis4.)