A262012 G.f.: [ Sum_{n>=0} (4*n)!/(n!)^4 * x^(4*n)/(1-x)^(4*n+4) ]^(1/4).

1, 1, 1, 1, 7, 31, 91, 211, 997, 5941, 27181, 97021, 369907, 1809211, 9180991, 40941031, 170062027, 753752971, 3645183691, 17473250251, 79444369189, 356738879533, 1662097580353, 7957682872873, 37696688946691, 175245959453491, 816849622436251, 3873243058472971, 18507865654295389

Offset

0,5

Links

Table of n, a(n) for n=0..28.

Formula

G.f. satisfies: A(x) = 1/(1-x) * Sum_{n>=0} A262013(n) * (x*A(x))^(4*n).

G.f.: A(x) = (1/x) * Series_Reversion( x / (G(x^4) + x) ) where G(x) is the g.f. of A262013.

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + 7*x^4 + 31*x^5 + 91*x^6 + 211*x^7 +...
such that
A(x)^4 = 1/(1-x)^4 + 24*x^4/(1-x)^8 + 2520*x^8/(1-x)^12 + 369600*x^12/(1-x)^16 + 63063000*x^16/(1-x)^20 + 11732745024*x^20/(1-x)^24 +...+ (4*n)!/(n!)^4*x^(4*n)/(1-x)^(4*n+4) +...
explicitly,
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 59*x^4 + 248*x^5 + 948*x^6 + 3000*x^7 + 10605*x^8 + 49468*x^9 + 238030*x^10 +...
Also, we have the series
x/Series_Reversion(x*A(x)) = 1+x + 6*x^4 + 432*x^8 + 45960*x^12 + 5780034*x^16 + 797957244*x^20 + 116916528960*x^24 + 17852845828752*x^28 + 2810058672255120*x^32 + 452703723158137776*x^36 + 74282858140993920000*x^40 +...+ A262013(n)*x^(4*n) +...
so that
A(x)*(1-x) = 1 + 6*x^4*A(x)^4 + 432*x^8*A(x)^8 + 45960*x^12*A(x)^12 + 5780034*x^16*A(x)^16 +...+ A262013(n)*(x*A(x))^(4*n) +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, x^(4*m)/(1-x +x*O(x^n))^(4*m+4)*(4*m)!/(m!)^4)^(1/4), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A262013.

Sequence in context: A164621 A202254 A305290 * A118934 A118935 A226838

Adjacent sequences: A262009 A262010 A262011 * A262013 A262014 A262015

Keyword

nonn

Author

Paul D. Hanna, Sep 11 2015

Status

approved