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%I #6 Apr 26 2019 20:45:28
%S 1,1,3,22,293,6396,192247,7487810,359234409,20571901624,1375291211051,
%T 105546986373918,9179888213024509,895591657896191348,
%U 97191443720659321503,11651115500493779746906,1533793016323134419714513,220607836736797491462201456,34514721658321036998293295955,5850764317548636197562977724758,1070825456345525996396825473813221
%N E.g.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n * exp(n^2*x).
%H Paul D. Hanna, <a href="/A325295/b325295.txt">Table of n, a(n) for n = 0..200</a>
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 293*x^4/4! + 6396*x^5/5! + 192247*x^6/6! + 7487810*x^7/7! + 359234409*x^8/8! + 20571901624*x^9/9! + 1375291211051*x^10/10! + ...
%e such that the following series are equal:
%e B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + x^9*A(x)^45 + ...
%e B(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) + x^5*exp(25*x) + x^6*exp(36*x) + x^7*exp(49*x) + x^8*exp(64*x) + x^9*exp(81*x) + ...
%e where
%e B(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! + 6622945*x^7/7! + 263313688*x^8/8! + ... + A193421(n)*x^n/n! + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A[#A] = -polcoeff( sum(m=0,#A, x^m*( Ser(A)^(m*(m+1)/2) - exp(m^2*x +x*O(x^#A))) ),#A) );n!*A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A193421.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 26 2019
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