A271932 G.f. A(x) satisfies: A(x) = A( x^7 + 7*x*A(x)^7 )^(1/7), with A(0)=0, A'(0)=1.

1, 1, 4, 20, 110, 638, 3828, 23515, 146970, 930820, 5957325, 38452405, 249944939, 1634287025, 10739831400, 70884562683, 469622328252, 3121694320866, 20811920304961, 139115729296575, 932107335003790, 6258662787526655, 42105353650697301, 283765005631661148, 1915495724241980280, 12949332513585521217, 87661142189041380207, 594176943178375193748, 4032121696383579351905, 27392082325012470506385, 186276500908841717917320

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 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146970*x^9 + 930820*x^10 + 5957325*x^11 + 38452405*x^12 +...
where A(x)^7 = A( x^7 + 7*x*A(x)^7 ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764822*x^15 + 40353901*x^16 + 282478679*x^17 + 1977362758*x^18 + 13841640148*x^19 + 96892304579*x^20 + 678252720401*x^21 +...
		

Links

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

Crossrefs

Cf. A271931, A271933.

Programs

PARI

{a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^7 + 7*X*A^7)^(1/7) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))