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

1, 4, 39, 604, 12625, 332766, 10574725, 393171416, 16744363569, 803841993370, 42957812253301, 2529951235854516, 162852898603253209, 11378885054925777494, 858009440175419213445, 69471138931959493061296, 6013997809048628612191585, 554545575488282609142617778

Offset: 1

Author

Paul D. Hanna, Mar 20 2023

Keywords

• nonn

Comments

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

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

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

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!.

Examples

			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! + ...
a(n) is divisible by n, where a(n)/n begins
[1, 2, 13, 151, 2525, 55461, 1510675, 49146427, 1860484841, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A361540, A185298.

Programs

PARI

/* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{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)}
for(n=1, 20, print1(A361540(n,1), ", "))