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A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n. 2
1, 8, 24, 0, -168, 0, 2112, 0, -32040, 0, 536256, 0, -9542976, 0, 177126912, 0, -3390361128, 0, 66436117440, 0, -1326185205696, 0, 26872637815296, 0, -551301904867392, 0, 11428295231789568, 0, -239010764560888320, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

G.f.: A(x) = x/Series_Reversion(x*G(x)^2)) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n = 1/agm(1, (1-16*x)^(1/2)) = g.f. of A002894 and G(x)^2 is the g.f. of A036917.

Self-convolution of A158101, which is a bisection of A158100; A158100 has g.f. F(x) that satisfies: F(x) = 1/AGM(1, 1 - 8*x/F(x) ).

a(n) = [x^n] AGM(1,(1-16x)^(1/2))^(2n-2)/(1-n) for n>1 where AGM is the arithmetic-geometric mean of Gauss. [From Paul D. Hanna, Mar 20 2010]

EXAMPLE

G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 +...

A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16x)^(1/2)) is the power series:

G(x) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 +...+ C(2n,n)^2*x^n +...

The square root of g.f. A(x) begins:

A(x)^(1/2) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 +...+ A158101(n)*x^n +...

PROG

(PARI) {a(n)=local(G=sum(m=0, n, binomial(2*m, m)^2*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G^2), n)}

(PARI) {a(n)=if(n==1, 8, polcoeff(agm(1, sqrt(1-16*x +x^2*O(x^n)))^(2*n-2), n)/(1-n))} [From Paul D. Hanna, Mar 20 2010]

CROSSREFS

Cf. A036917, A002894, A158101, A158100.

Sequence in context: A036562 A220973 A233470 * A250035 A263630 A205376

Adjacent sequences:  A172387 A172388 A172389 * A172391 A172392 A172393

KEYWORD

sign

AUTHOR

Paul D. Hanna, Feb 04 2010

STATUS

approved

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Last modified November 21 07:06 EST 2017. Contains 294992 sequences.