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a(n) = A361540(n, n-2) for n >= 2, a diagonal of triangle A361540.
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%I #7 Mar 24 2023 09:04:03

%S 3,39,426,4550,50085,577731,7022596,90148860,1222753815,17515226465,

%T 264663151038,4212100028994,70475063838361,1237144088015535,

%U 22735980569119560,436467520716475064,8733235757816095083,181740089309026259565,3925458146197916823970

%N a(n) = A361540(n, n-2) for n >= 2, a diagonal of triangle A361540.

%C E.g.f. F(x,y) of triangle A361540 satisfies the following.

%C (1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.

%C (2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.

%C The diagonal above this one in triangle A361540 has e.g.f. x*exp(x)*exp(x*exp(x)).

%H Paul D. Hanna, <a href="/A361539/b361539.txt">Table of n, a(n) for n = 2..42</a>

%e E.g.f. A(x) = 3*x^2/2! + 39*x^3/3! + 426*x^4/4! + 4550*x^5/5! + 50085*x^6/6! + 577731*x^7/7! + 7022596*x^8/8! + 90148860*x^9/9! + 1222753815*x^10/10! + ...

%o (PARI) /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */

%o {A361540(n,k) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(A, n,x),k,y)}

%o for(n=2, 20, print1(A361540(n,n-2), ", "))

%Y Cf. A361540.

%K nonn

%O 2,1

%A _Paul D. Hanna_, Mar 20 2023