A362718 Expansion of e.g.f. cos(x)*exp(x^2/2) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.

1, 0, -2, -16, -132, -1216, -12440, -138048, -1601264, -18108928, -161934624, 404007680, 92590134208, 4221314202624, 159324751301248, 5730872535686144, 205239818509082880, 7450322829180649472, 276342876017093172736, 10509280308463090102272

Offset: 0

Author

Michael Somos, Apr 30 2023

Keywords

• sign

Crossrefs

Cf. A001464.

Programs

Mathematica

a[ n_] := If[ n<0, 0, (2*n)! * SeriesCoefficient[ Cos[x] * Exp[x^2/2], {x, 0, 2*n}]];

PARI

{a(n) = my(A); if( n<0, 0, A = x*O(x^(2*n)); (2*n)! * polcoef( cos(x + A)*exp(x^2/2 + A), 2*n))};

SageMath

def egfExpand(f, step, size) -> list[int]:
    x = LazyPowerSeriesRing(QQ, "x").gen()
    return [f(x)[step*n] * factorial(step*n) for n in range(size+1)]
def egf(x): return cos(x)*exp(x^2/2)
print(egfExpand(egf, 2, 19))  # Peter Luschny, May 02 2023

Formula

a(n) = (-1)^n * A001464(2*n).

0 = a(n)*(360*a(n+2) -600*a(n+3) +230*a(n+4) -28*a(n+5) +a(n+6)) +a(n+1)*(216*a(n+2) -296*a(n+3) +84*a(n+4) -6*a(n+5)) +a(n+2)*(66*a(n+2) -56*a(n+3) +15*a(n+4)) -10*a(n+3)^2 for all n >= 0.