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A171802 G.f. satisfies: A(x) = P(x*A(x)^2) where A(x/P(x)^2) = P(x) is the g.f. for Partition numbers (A000041). 4
1, 1, 4, 20, 115, 714, 4669, 31671, 220800, 1572395, 11389059, 83642650, 621400794, 4661706035, 35264616260, 268700873765, 2060348179869, 15886552304352, 123102352038195, 958128272163860, 7487015421267228, 58715989507106041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

FORMULA

G.f. A(x) satisfies [Paul D. Hanna, Nov 24 2012]:

(1) A(x) = (1/x)*series_reversion(x*eta(x)^2).

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 8.42516721063251541777601555584151410936... and c = 0.2128745515668564974075326286129891378270... - Vaclav Kotesovec, May 13 2018

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 +...

G.f. satisfies A(x/P(x)^2) = P(x) where:

P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...

and x/P(x)^2 = x - 2*x^2 - x^3 + 2*x^4 + x^5 + 2*x^6 - 2*x^7 - 2*x^9 +...

PROG

(PARI) a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^2))^(1/2), n)

CROSSREFS

Cf. A171803, A171804, A171805, A109085, A000041, A304444.

Sequence in context: A320615 A316298 A291531 * A100034 A192924 A258664

Adjacent sequences:  A171799 A171800 A171801 * A171803 A171804 A171805

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 19 2009

STATUS

approved

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Last modified August 8 13:43 EDT 2020. Contains 336298 sequences. (Running on oeis4.)