cp's OEIS Frontend

A361544 a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.

%I A361544 #7 Mar 24 2023 09:03:42
%S A361544 1,4,39,604,12625,332766,10574725,393171416,16744363569,803841993370,
%T A361544 42957812253301,2529951235854516,162852898603253209,
%U A361544 11378885054925777494,858009440175419213445,69471138931959493061296,6013997809048628612191585,554545575488282609142617778
%N A361544 a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.
%C A361544 E.g.f. F(x,y) of triangle A361540 satisfies the following.
%C A361544 (1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
%C A361544 (2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
%C A361544 The column next to this one in triangle A361540 has e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.
%H A361544 Paul D. Hanna, <a href="/A361544/b361544.txt">Table of n, a(n) for n = 1..42</a>
%e A361544 E.g.f.: A(x) = x + 4*x^2/2! + 39*x^3/3! + 604*x^4/4! + 12625*x^5/5! + 332766*x^6/6! + 10574725*x^7/7! + 393171416*x^8/8! + 16744363569*x^9/9! + 803841993370*x^10/10! + ... + a(n)*x^n/n! + ...
%e A361544 a(n) is divisible by n, where a(n)/n begins
%e A361544 [1, 2, 13, 151, 2525, 55461, 1510675, 49146427, 1860484841, ...].
%o A361544 (PARI) /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
%o A361544 {A361540(n,k) = my(F = 1); for(i=1,n, F = sum(m=0, n, (F^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(F, n,x),k,y)}
%o A361544 for(n=1, 20, print1(A361540(n,1), ", "))
%Y A361544 Cf. A361540, A185298.
%K A361544 nonn
%O A361544 1,2
%A A361544 _Paul D. Hanna_, Mar 20 2023