A369746 Expansion of e.g.f. exp( 3 * (1-sqrt(1-2*x)) ).

1, 3, 12, 63, 423, 3528, 35559, 422901, 5817744, 91072269, 1600588269, 31230827532, 670252672593, 15696888917427, 398454496989012, 10899543418960167, 319672849622745951, 10007954229075765984, 333139545206104991031, 11749955670275356579941

Offset: 0

Author

Seiichi Manyama, Jan 30 2024

Keywords

• nonn

Crossrefs

Cf. A107104, A144301, A132062.

Cf. A369722.

Programs

Maple

# The row polynomials of A132062 evaluated at x = 3.
T := proc(n, k) option remember; if k = 0 then 0^n elif n < k then 0
else (2*(n - 1) - k)*T(n - 1, k) + T(n - 1, k - 1) fi end:
seq(add(T(n, k)*3^k, k = 0..n), n = 0..19);  # Peter Luschny, Apr 25 2024

Mathematica

With[{nn=20},CoefficientList[Series[Exp[3(1-Sqrt[1-2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(1-sqrt(1-2*x)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^(n-k) * (n-1+k)! / (2^k * k! * (n-1-k)!).

a(n) = (2*n-3)*a(n-1) + 9*a(n-2).