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

1,2

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!.

Links

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

Example

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, ...].

Prog

(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), ", "))

Crossrefs

Cf. A361540, A185298.

Sequence in context: A192935 A365010 A024055 * A178920 A066399 A065760

Adjacent sequences: A361541 A361542 A361543 * A361545 A361546 A361547

Keyword

nonn

Author

Paul D. Hanna, Mar 20 2023

Status

approved