A298693 - OEIS

%I #7 Feb 09 2018 03:25:54

%S 1,1,3,22,294,5911,158293,5251690,206696194,9387611937,482745371458,

%T 27717788095397,1757818683339028,122058148921357056,

%U 9212494564360610855,751138761646263512978,65807775099574132000968,6166278653572358495161057,615421469545011786309942067,65183859793912213778457542207,7303117991652113167690085149033

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^( n*(n+1) ).

%e G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 294*x^4 + 5911*x^5 + 158293*x^6 + 5251690*x^7 + 206696194*x^8 + 9387611937*x^9 + 482745371458*x^10 + 27717788095397*x^11 + 1757818683339028*x^12 + 122058148921357056*x^13 + 9212494564360610855*x^14 + 751138761646263512978*x^15 + ...

%e such that

%e A(x) = 1 + C(2,1)/2*x/A(x)^2 + C(6,2)/3*x^2/A(x)^6 + C(12,3)/4*x^3/A(x)^12 + C(20,4)/5*x^4/A(x)^20 + C(30,5)/6*x^5/A(x)^30 + C(42,6)/7*x^6/A(x)^42 + C(56,7)/8*x^7/A(x)^56 + ...

%e more explicitly,

%e A(x) = 1 + x/A(x)^2 + 5*x^2/A(x)^6 + 55*x^3/A(x)^12 + 969*x^4/A(x)^20 + 23751*x^5/A(x)^30 + 749398*x^6/A(x)^42 + 28989675*x^7/A(x)^56 + ... + A135861(n)*x^n/A(x)^(n*(n+1)) + ...

%t terms = 21; A[_] = 1; Do[A[x_] = 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/ A[x]^(n*(n+1)), {n, terms}] + O[x]^terms, {terms}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Feb 09 2018 *)

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1),m)/(m+1) * x^m/Ser(A)^(m*(m+1)) )));  A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A298692, A135861.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 03 2018