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A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041). 4
1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;

(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).

G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Self-convolution cube of A171804 (with offset).

a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017

EXAMPLE

G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...

where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and

x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...

MATHEMATICA

InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

PROG

(PARI) {a(n)=polcoeff(serreverse(x*eta(x+x*O(x^n))^3), n)}

CROSSREFS

Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.

Sequence in context: A120922 A185113 A291775 * A154931 A047731 A291841

Adjacent sequences:  A171802 A171803 A171804 * A171806 A171807 A171808

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 20 2009

EXTENSIONS

More terms from Vladimir Reshetnikov, Nov 21 2016

STATUS

approved

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Last modified July 10 10:17 EDT 2020. Contains 335576 sequences. (Running on oeis4.)