|
%I #9 Feb 09 2018 03:26:01
%S 1,1,4,32,419,8052,207784,6724274,260396693,11697865930,596886780272,
%T 34072732137625,2151062784054901,148819021611467291,
%U 11198412956841549966,910736443741061568539,79616310026220269203631,7446056807577515910468813,741918566779386113373532994,78467177619239380045368550016,8779922184077661414128958823323
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^(n^2).
%e G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 419*x^4 + 8052*x^5 + 207784*x^6 + 6724274*x^7 + 260396693*x^8 + 11697865930*x^9 + 596886780272*x^10 + 34072732137625*x^11 + 2151062784054901*x^12 + 148819021611467291*x^13 + 11198412956841549966*x^14 + 910736443741061568539*x^15 + ...
%e such that
%e A(x) = 1 + C(2,1)/2*x/A(x) + C(6,2)/3*x^2/A(x)^4 + C(12,3)/4*x^3/A(x)^9 + C(20,4)/5*x^4/A(x)^16 + C(30,5)/6*x^5/A(x)^25 + C(42,6)/7*x^6/A(x)^36 + C(56,7)/8*x^7/A(x)^49 + ...
%e more explicitly,
%e A(x) = 1 + x/A(x) + 5*x^2/A(x)^4 + 55*x^3/A(x)^9 + 969*x^4/A(x)^16 + 23751*x^5/A(x)^25 + 749398*x^6/A(x)^36 + 28989675*x^7/A(x)^49 + ... + A135861(n)*x^n/A(x)^(n^2) + ...
%t terms = 21; A[_] = 1; Do[A[x_] = 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/ A[x]^(n^2), {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^2) ))); A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A298691, A298692, A298693, A135861.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 03 2018
|
