A271933 G.f. A(x) satisfies: A(x) = A( x^11 + 11*x*A(x)^11 )^(1/11), with A(0)=0, A'(0)=1.

1, 1, 6, 46, 391, 3519, 32844, 314364, 3065049, 30309929, 303099290, 3058547381, 31095231708, 318128139796, 3272175152355, 33812476576290, 350804444501589, 3652493334187197, 38148263715573364, 399552867370295155, 4195305107766973240, 44150591852677070280, 465588059585378099226, 4919039064854516328821, 52059830109088065802395, 551834199223958450647359, 5857932269440676202573084

Offset: 1

Author

Paul D. Hanna, Apr 16 2016

Keywords

• nonn

Comments

Compare the g.f. to the following identities:

(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),

(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),

where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 46*x^4 + 391*x^5 + 3519*x^6 + 32844*x^7 + 314364*x^8 + 3065049*x^9 + 30309929*x^10 + 303099290*x^11 + 3058547381*x^12 +...
where A(x)^11 = A( x^11 + 11*x*A(x)^11 ).
RELATED SERIES.
A(x)^11 = x^11 + 11*x^12 + 121*x^13 + 1331*x^14 + 14641*x^15 + 161051*x^16 + 1771561*x^17 + 19487171*x^18 + 214358881*x^19 + 2357947691*x^20 + 25937424601*x^21 + 285311670612*x^22 + 3138428376754*x^23 + 34522712144657*x^24 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A271931, A271932.

Programs

PARI

{a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^11 + 11*X*A^11)^(1/11) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))