A361549 a(n) = A361540(n,2) for n >= 2, a column of triangle A361540.

1, 18, 426, 12040, 401355, 15456756, 676130644, 33151425840, 1802216703285, 107652497473180, 7012494336544686, 494963689847333928, 37648456802884402111, 3071415347513049808740, 267644521958509484952360, 24822151072519637091258976, 2442314922307988498911793385

Offset: 2

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!.

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!.

Examples

			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! + ...
a(n) is divisible by n*(n-1)/2, where a(n)*2/(n*(n-1)) begins
[1, 6, 71, 1204, 26757, 736036, 24147523, 920872940, 40049260073, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 2..42

Crossrefs

Cf. A361540, A361544.

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