A361549 - OEIS

%I #7 Mar 24 2023 09:03:48

%S 1,18,426,12040,401355,15456756,676130644,33151425840,1802216703285,

%T 107652497473180,7012494336544686,494963689847333928,

%U 37648456802884402111,3071415347513049808740,267644521958509484952360,24822151072519637091258976,2442314922307988498911793385

%N a(n) = A361540(n,2) for n >= 2, a column 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 Column 0 near to this one in triangle A361540 has e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.

%H Paul D. Hanna, <a href="/A361549/b361549.txt">Table of n, a(n) for n = 2..42</a>

%e E.g.f.: A(x) = x^2/2! + 18*x^3/3! + 426*x^4/4! + 12040*x^5/5! + 401355*x^6/6! + 15456756*x^7/7! + 676130644*x^8/8! + 33151425840*x^9/9! + 1802216703285*x^10/10! + ... + a(n)*x^n/n! + ...

%e a(n) is divisible by n*(n-1)/2, where a(n)*2/(n*(n-1)) begins

%e [1, 6, 71, 1204, 26757, 736036, 24147523, 920872940, 40049260073, ...].

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

%Y Cf. A361540, A361544.

%K nonn

%O 2,2

%A _Paul D. Hanna_, Mar 20 2023