A172393 - OEIS

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A172393

G.f. satisfies: A(x) = G(x/A(x)^2) and G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n,n)*C(2n+2,n+1)/(n+2)*x^n is the g.f. of A172392.

1, 4, -2, 8, -20, 96, -324, 1648, -6348, 33200, -137848, 732640, -3193296, 17148608, -77335400, 418289696, -1934677436, 10518803376, -49611450120, 270796872160, -1297234193744, 7102371571840, -34458382484976, 189117499963840

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..23.

FORMULA

G.f. satisfies: A(x) = Sum_{n>=0} A000108(n+1)*A000984(n)*x^n/A(x)^(2n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.

Self-convolution equals A172391.

EXAMPLE

G.f.: A(x) = 1 + 4*x - 2*x^2 + 8*x^3 - 20*x^4 + 96*x^5 - 324*x^6 +...

A(x)^2 = 1 + 8*x + 12*x^2 + 28*x^4 + 264*x^6 + 3720*x^8 +...

where A(x)^2 equals the g.f. of A172391:

A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].

Let G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n+2,n+1)/(n+2)*C(2n,n)*x^n:

G(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...

PROG

(PARI) {a(n)=local(G=sum(m=0, n, binomial(2*m, m)*binomial(2*m+2, m+1)/(m+2)*x^m)+x*O(x^n)); polcoeff((x/serreverse(x*G^2))^(1/2), n)}

CROSSREFS

Cf. A172391, A172392.