A360348 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for n >= 1.

1, 9, 100, 1381, 22771, 435138, 9442049, 229265109, 6160375990, 181559237499, 5826147967201, 202295647539886, 7559401377952659, 302570522540568557, 12917629672442043340, 586047019821392518293, 28159186576616423049683, 1428679795354280280080736, 76329278834398327082152543

Offset

1,2

Comments

Related sequence: A002426(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) ~ BesselI(0, 2) * n^(n+1). - Vaclav Kotesovec, Feb 12 2023

Example

L.g.f.: A(x) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + 6160375990*x^9/9 + ...
a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + (y^12 + 66*y^11 + 1491*y^10 + 16002*y^9 + 91293*y^8 + 281220*y^7 + 435138*y^6 + 281220*y^5 + 91293*y^4 + 16002*y^3 + 1491*y^2 + 66*y + 1)*x^6/6 + ...
Exponentiation yields the g.f. of A360349:
exp(A(x)) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + ... + A360349(n)*x^n + ...

Prog

(PARI) {a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A360349, A360238, A002426.

Sequence in context: A202833 A356131 A287039 * A174509 A103461 A101563

Adjacent sequences: A360345 A360346 A360347 * A360349 A360350 A360351

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2023

Status

approved